# SpectralSequence ^ ZZ -- the kth page of a spectral sequence

## Synopsis

• Usage:
P = E^k
• Operator: ^
• Inputs:
• Outputs:
• SpectralSequencePage =>

## Description

Returns the kth page of the spectral sequence.

Consider the filtered complex K below, obtained by multiplying the minimal free resolution of the rational quartic space curve by sucessive powers of the irrelevant ideal.

 `i1 : B = QQ[a..d];` ```i2 : J = ideal vars B; o2 : Ideal of B``` `i3 : C = complete res monomialCurveIdeal(B,{1,3,4});` `i4 : K = filteredComplex(J,C,4);`

Let E be the spectral sequence determined by K.

 `i5 : E = spectralSequence K;`

We now compute some pages.

 ```i6 : E^0 +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------+ o6 = |image | a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 0 0 0 0 0 ||subquotient ({2} | a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 -a4b2 -a5c -a4bd -a4c2 -a3b3 -a4bc -a3b2d -a3bc2 -a3b2c -a4c2 -a3bcd -a3c3 -a3b2d -a4cd -a3bd2 -a3c2d -a2b4 -a3b2c -a2b3d -a2b2c2 -a2b3c -a3bc2 -a2b2cd -a2bc3 -a2b3d -a3bcd -a2b2d2 -a2bc2d -a2b2c2 -a3c3 -a2bc2d -a2c4 -a2b2cd -a3c2d -a2bcd2 -a2c3d -a2b2d2 -a3cd2 -a2bd3 -a2c2d2 -ab5 -a2b3c -ab4d -ab3c2 -ab4c -a2b2c2 -ab3cd -ab2c3 -ab4d -a2b2cd -ab3d2 -ab2c2d -ab3c2 -a2bc3 -ab2c2d -abc4 -ab3cd -a2bc2d -ab2cd2 -abc3d -ab3d2 -a2bcd2 -ab2d3 -abc2d2 -ab2c3 -a2c4 -abc3d -ac5 -ab2c2d -a2c3d -abc2d2 -ac4d -ab2cd2 -a2c2d2 -abcd3 -ac3d2 -ab2d3 -a2cd3 -abd4 -ac2d3 -b6 -ab4c -b5d -b4c2 -b5c -ab3c2 -b4cd -b3c3 -b5d -ab3cd -b4d2 -b3c2d -b4c2 -ab2c3 -b3c2d -b2c4 -b4cd -ab2c2d -b3cd2 -b2c3d -b4d2 -ab2cd2 -b3d3 -b2c2d2 -b3c3 -abc4 -b2c3d -bc5 -b3c2d -abc3d -b2c2d2 -bc4d -b3cd2 -abc2d2 -b2cd3 -bc3d2 -b3d3 -abcd3 -b2d4 -bc2d3 -b2c4 -ac5 -bc4d -c6 -b2c3d -ac4d -bc3d2 -c5d -b2c2d2 -ac3d2 -bc2d3 -c4d2 -b2cd3 -ac2d3 -bcd4 -c3d3 -b2d4 -acd4 -bd5 -c2d4 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 |, {2} | -a4b2 -a5c -a4bd -a4c2 -a3b3 -a4bc -a3b2d -a3bc2 -a3b2c -a4c2 -a3bcd -a3c3 -a3b2d -a4cd -a3bd2 -a3c2d -a2b4 -a3b2c -a2b3d -a2b2c2 -a2b3c -a3bc2 -a2b2cd -a2bc3 -a2b3d -a3bcd -a2b2d2 -a2bc2d -a2b2c2 -a3c3 -a2bc2d -a2c4 -a2b2cd -a3c2d -a2bcd2 -a2c3d -a2b2d2 -a3cd2 -a2bd3 -a2c2d2 -ab5 -a2b3c -ab4d -ab3c2 -ab4c -a2b2c2 -ab3cd -ab2c3 -ab4d -a2b2cd -ab3d2 -ab2c2d -ab3c2 -a2bc3 -ab2c2d -abc4 -ab3cd -a2bc2d -ab2cd2 -abc3d -ab3d2 -a2bcd2 -ab2d3 -abc2d2 -ab2c3 -a2c4 -abc3d -ac5 -ab2c2d -a2c3d -abc2d2 -ac4d -ab2cd2 -a2c2d2 -abcd3 -ac3d2 -ab2d3 -a2cd3 -abd4 -ac2d3 -b6 -ab4c -b5d -b4c2 -b5c -ab3c2 -b4cd -b3c3 -b5d -ab3cd -b4d2 -b3c2d -b4c2 -ab2c3 -b3c2d -b2c4 -b4cd -ab2c2d -b3cd2 -b2c3d -b4d2 -ab2cd2 -b3d3 -b2c2d2 -b3c3 -abc4 -b2c3d -bc5 -b3c2d -abc3d -b2c2d2 -bc4d -b3cd2 -abc2d2 -b2cd3 -bc3d2 -b3d3 -abcd3 -b2d4 -bc2d3 -b2c4 -ac5 -bc4d -c6 -b2c3d -ac4d -bc3d2 -c5d -b2c2d2 -ac3d2 -bc2d3 -c4d2 -b2cd3 -ac2d3 -bcd4 -c3d3 -b2d4 -acd4 -bd5 -c2d4 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 |) |subquotient ({4} | a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 a3d a2bd a2cd a2d2 ab2d abcd abd2 ac2d acd2 ad3 b3d b2cd b2d2 bc2d bcd2 bd3 c3d c2d2 cd3 d4 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |, {4} | a3d a2bd a2cd a2d2 ab2d abcd abd2 ac2d acd2 ad3 b3d b2cd b2d2 bc2d bcd2 bd3 c3d c2d2 cd3 d4 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |) |subquotient ({5} | a b c d a2 ab ac ad b2 bc bd c2 cd d2 |, {5} | a2 ab ac ad b2 bc bd c2 cd d2 |) |0 | | | {3} | 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 a4c a4d 0 0 a3bc a3bd 0 0 a3c2 a3cd 0 0 a3cd a3d2 0 0 a2b2c a2b2d 0 0 a2bc2 a2bcd 0 0 a2bcd a2bd2 0 0 a2c3 a2c2d 0 0 a2c2d a2cd2 0 0 a2cd2 a2d3 0 0 ab3c ab3d 0 0 ab2c2 ab2cd 0 0 ab2cd ab2d2 0 0 abc3 abc2d 0 0 abc2d abcd2 0 0 abcd2 abd3 0 0 ac4 ac3d 0 0 ac3d ac2d2 0 0 ac2d2 acd3 0 0 acd3 ad4 0 0 b4c b4d 0 0 b3c2 b3cd 0 0 b3cd b3d2 0 0 b2c3 b2c2d 0 0 b2c2d b2cd2 0 0 b2cd2 b2d3 0 0 bc4 bc3d 0 0 bc3d bc2d2 0 0 bc2d2 bcd3 0 0 bcd3 bd4 0 0 c5 c4d 0 0 c4d c3d2 0 0 c3d2 c2d3 0 0 c2d3 cd4 0 0 cd4 d5 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 | {3} | a4c a4d 0 0 a3bc a3bd 0 0 a3c2 a3cd 0 0 a3cd a3d2 0 0 a2b2c a2b2d 0 0 a2bc2 a2bcd 0 0 a2bcd a2bd2 0 0 a2c3 a2c2d 0 0 a2c2d a2cd2 0 0 a2cd2 a2d3 0 0 ab3c ab3d 0 0 ab2c2 ab2cd 0 0 ab2cd ab2d2 0 0 abc3 abc2d 0 0 abc2d abcd2 0 0 abcd2 abd3 0 0 ac4 ac3d 0 0 ac3d ac2d2 0 0 ac2d2 acd3 0 0 acd3 ad4 0 0 b4c b4d 0 0 b3c2 b3cd 0 0 b3cd b3d2 0 0 b2c3 b2c2d 0 0 b2c2d b2cd2 0 0 b2cd2 b2d3 0 0 bc4 bc3d 0 0 bc3d bc2d2 0 0 bc2d2 bcd3 0 0 bcd3 bd4 0 0 c5 c4d 0 0 c4d c3d2 0 0 c3d2 c2d3 0 0 c2d3 cd4 0 0 cd4 d5 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 | | {4} | 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 -a3c -a2bc -a2c2 -a2cd -ab2c -abc2 -abcd -ac3 -ac2d -acd2 -b3c -b2c2 -b2cd -bc3 -bc2d -bcd2 -c4 -c3d -c2d2 -cd3 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | {4} | -a3c -a2bc -a2c2 -a2cd -ab2c -abc2 -abcd -ac3 -ac2d -acd2 -b3c -b2c2 -b2cd -bc3 -bc2d -bcd2 -c4 -c3d -c2d2 -cd3 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | | | | |{-4, 4} | {3} | 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 a5 a4b -a4c -a4d a4b a3b2 -a3bc -a3bd a4c a3bc -a3c2 -a3cd a4d a3bd -a3cd -a3d2 a3b2 a2b3 -a2b2c -a2b2d a3bc a2b2c -a2bc2 -a2bcd a3bd a2b2d -a2bcd -a2bd2 a3c2 a2bc2 -a2c3 -a2c2d a3cd a2bcd -a2c2d -a2cd2 a3d2 a2bd2 -a2cd2 -a2d3 a2b3 ab4 -ab3c -ab3d a2b2c ab3c -ab2c2 -ab2cd a2b2d ab3d -ab2cd -ab2d2 a2bc2 ab2c2 -abc3 -abc2d a2bcd ab2cd -abc2d -abcd2 a2bd2 ab2d2 -abcd2 -abd3 a2c3 abc3 -ac4 -ac3d a2c2d abc2d -ac3d -ac2d2 a2cd2 abcd2 -ac2d2 -acd3 a2d3 abd3 -acd3 -ad4 ab4 b5 -b4c -b4d ab3c b4c -b3c2 -b3cd ab3d b4d -b3cd -b3d2 ab2c2 b3c2 -b2c3 -b2c2d ab2cd b3cd -b2c2d -b2cd2 ab2d2 b3d2 -b2cd2 -b2d3 abc3 b2c3 -bc4 -bc3d abc2d b2c2d -bc3d -bc2d2 abcd2 b2cd2 -bc2d2 -bcd3 abd3 b2d3 -bcd3 -bd4 ac4 bc4 -c5 -c4d ac3d bc3d -c4d -c3d2 ac2d2 bc2d2 -c3d2 -c2d3 acd3 bcd3 -c2d3 -cd4 ad4 bd4 -cd4 -d5 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | {3} | a5 a4b -a4c -a4d a4b a3b2 -a3bc -a3bd a4c a3bc -a3c2 -a3cd a4d a3bd -a3cd -a3d2 a3b2 a2b3 -a2b2c -a2b2d a3bc a2b2c -a2bc2 -a2bcd a3bd a2b2d -a2bcd -a2bd2 a3c2 a2bc2 -a2c3 -a2c2d a3cd a2bcd -a2c2d -a2cd2 a3d2 a2bd2 -a2cd2 -a2d3 a2b3 ab4 -ab3c -ab3d a2b2c ab3c -ab2c2 -ab2cd a2b2d ab3d -ab2cd -ab2d2 a2bc2 ab2c2 -abc3 -abc2d a2bcd ab2cd -abc2d -abcd2 a2bd2 ab2d2 -abcd2 -abd3 a2c3 abc3 -ac4 -ac3d a2c2d abc2d -ac3d -ac2d2 a2cd2 abcd2 -ac2d2 -acd3 a2d3 abd3 -acd3 -ad4 ab4 b5 -b4c -b4d ab3c b4c -b3c2 -b3cd ab3d b4d -b3cd -b3d2 ab2c2 b3c2 -b2c3 -b2c2d ab2cd b3cd -b2c2d -b2cd2 ab2d2 b3d2 -b2cd2 -b2d3 abc3 b2c3 -bc4 -bc3d abc2d b2c2d -bc3d -bc2d2 abcd2 b2cd2 -bc2d2 -bcd3 abd3 b2d3 -bcd3 -bd4 ac4 bc4 -c5 -c4d ac3d bc3d -c4d -c3d2 ac2d2 bc2d2 -c3d2 -c2d3 acd3 bcd3 -c2d3 -cd4 ad4 bd4 -cd4 -d5 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | | {4} | 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 -a3b -a2b2 -a2bc -a2bd -ab3 -ab2c -ab2d -abc2 -abcd -abd2 -b4 -b3c -b3d -b2c2 -b2cd -b2d2 -bc3 -bc2d -bcd2 -bd3 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | {4} | -a3b -a2b2 -a2bc -a2bd -ab3 -ab2c -ab2d -abc2 -abcd -abd2 -b4 -b3c -b3d -b2c2 -b2cd -b2d2 -bc3 -bc2d -bcd2 -bd3 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | |{-1, 4} |{0, 4} | | | {3} | 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 a5 a4b 0 0 a4b a3b2 0 0 a4c a3bc 0 0 a4d a3bd 0 0 a3b2 a2b3 0 0 a3bc a2b2c 0 0 a3bd a2b2d 0 0 a3c2 a2bc2 0 0 a3cd a2bcd 0 0 a3d2 a2bd2 0 0 a2b3 ab4 0 0 a2b2c ab3c 0 0 a2b2d ab3d 0 0 a2bc2 ab2c2 0 0 a2bcd ab2cd 0 0 a2bd2 ab2d2 0 0 a2c3 abc3 0 0 a2c2d abc2d 0 0 a2cd2 abcd2 0 0 a2d3 abd3 0 0 ab4 b5 0 0 ab3c b4c 0 0 ab3d b4d 0 0 ab2c2 b3c2 0 0 ab2cd b3cd 0 0 ab2d2 b3d2 0 0 abc3 b2c3 0 0 abc2d b2c2d 0 0 abcd2 b2cd2 0 0 abd3 b2d3 0 0 ac4 bc4 0 0 ac3d bc3d 0 0 ac2d2 bc2d2 0 0 acd3 bcd3 0 0 ad4 bd4 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 | {3} | 0 0 a5 a4b 0 0 a4b a3b2 0 0 a4c a3bc 0 0 a4d a3bd 0 0 a3b2 a2b3 0 0 a3bc a2b2c 0 0 a3bd a2b2d 0 0 a3c2 a2bc2 0 0 a3cd a2bcd 0 0 a3d2 a2bd2 0 0 a2b3 ab4 0 0 a2b2c ab3c 0 0 a2b2d ab3d 0 0 a2bc2 ab2c2 0 0 a2bcd ab2cd 0 0 a2bd2 ab2d2 0 0 a2c3 abc3 0 0 a2c2d abc2d 0 0 a2cd2 abcd2 0 0 a2d3 abd3 0 0 ab4 b5 0 0 ab3c b4c 0 0 ab3d b4d 0 0 ab2c2 b3c2 0 0 ab2cd b3cd 0 0 ab2d2 b3d2 0 0 abc3 b2c3 0 0 abc2d b2c2d 0 0 abcd2 b2cd2 0 0 abd3 b2d3 0 0 ac4 bc4 0 0 ac3d bc3d 0 0 ac2d2 bc2d2 0 0 acd3 bcd3 0 0 ad4 bd4 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 | | {4} | 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | {4} | a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | | | | | | | | | | | |{-3, 4} |{-2, 4} | | | 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|0 |subquotient (| a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 a4bc-a5d a4b3-a6c a5c2-a4b2d a4c3-a4bd2 a3b2c-a4bd a3b4-a5bc a4bc2-a3b3d a3bc3-a3b2d2 a3bc2-a4cd a3b3c-a5c2 a4c3-a3b2cd a3c4-a3bcd2 a3bcd-a4d2 a3b3d-a5cd a4c2d-a3b2d2 a3c3d-a3bd3 a2b3c-a3b2d a2b5-a4b2c a3b2c2-a2b4d a2b2c3-a2b3d2 a2b2c2-a3bcd a2b4c-a4bc2 a3bc3-a2b3cd a2bc4-a2b2cd2 a2b2cd-a3bd2 a2b4d-a4bcd a3bc2d-a2b3d2 a2bc3d-a2b2d3 a2bc3-a3c2d a2b3c2-a4c3 a3c4-a2b2c2d a2c5-a2bc2d2 a2bc2d-a3cd2 a2b3cd-a4c2d a3c3d-a2b2cd2 a2c4d-a2bcd3 a2bcd2-a3d3 a2b3d2-a4cd2 a3c2d2-a2b2d3 a2c3d2-a2bd4 ab4c-a2b3d ab6-a3b3c a2b3c2-ab5d ab3c3-ab4d2 ab3c2-a2b2cd ab5c-a3b2c2 a2b2c3-ab4cd ab2c4-ab3cd2 ab3cd-a2b2d2 ab5d-a3b2cd a2b2c2d-ab4d2 ab2c3d-ab3d3 ab2c3-a2bc2d ab4c2-a3bc3 a2bc4-ab3c2d abc5-ab2c2d2 ab2c2d-a2bcd2 ab4cd-a3bc2d a2bc3d-ab3cd2 abc4d-ab2cd3 ab2cd2-a2bd3 ab4d2-a3bcd2 a2bc2d2-ab3d3 abc3d2-ab2d4 abc4-a2c3d ab3c3-a3c4 a2c5-ab2c3d ac6-abc3d2 abc3d-a2c2d2 ab3c2d-a3c3d a2c4d-ab2c2d2 ac5d-abc2d3 abc2d2-a2cd3 ab3cd2-a3c2d2 a2c3d2-ab2cd3 ac4d2-abcd4 abcd3-a2d4 ab3d3-a3cd3 a2c2d3-ab2d4 ac3d3-abd5 b5c-ab4d b7-a2b4c ab4c2-b6d b4c3-b5d2 b4c2-ab3cd b6c-a2b3c2 ab3c3-b5cd b3c4-b4cd2 b4cd-ab3d2 b6d-a2b3cd ab3c2d-b5d2 b3c3d-b4d3 b3c3-ab2c2d b5c2-a2b2c3 ab2c4-b4c2d b2c5-b3c2d2 b3c2d-ab2cd2 b5cd-a2b2c2d ab2c3d-b4cd2 b2c4d-b3cd3 b3cd2-ab2d3 b5d2-a2b2cd2 ab2c2d2-b4d3 b2c3d2-b3d4 b2c4-abc3d b4c3-a2bc4 abc5-b3c3d bc6-b2c3d2 b2c3d-abc2d2 b4c2d-a2bc3d abc4d-b3c2d2 bc5d-b2c2d3 b2c2d2-abcd3 b4cd2-a2bc2d2 abc3d2-b3cd3 bc4d2-b2cd4 b2cd3-abd4 b4d3-a2bcd3 abc2d3-b3d4 bc3d3-b2d5 bc5-ac4d b3c4-a2c5 ac6-b2c4d c7-bc4d2 bc4d-ac3d2 b3c3d-a2c4d ac5d-b2c3d2 c6d-bc3d3 bc3d2-ac2d3 b3c2d2-a2c3d2 ac4d2-b2c2d3 c5d2-bc2d4 bc2d3-acd4 b3cd3-a2c2d3 ac3d3-b2cd4 c4d3-bcd5 bcd4-ad5 b3d4-a2cd4 ac2d4-b2d5 c3d4-bd6 a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 |, | a4bc-a5d a4b3-a6c a5c2-a4b2d a4c3-a4bd2 a3b2c-a4bd a3b4-a5bc a4bc2-a3b3d a3bc3-a3b2d2 a3bc2-a4cd a3b3c-a5c2 a4c3-a3b2cd a3c4-a3bcd2 a3bcd-a4d2 a3b3d-a5cd a4c2d-a3b2d2 a3c3d-a3bd3 a2b3c-a3b2d a2b5-a4b2c a3b2c2-a2b4d a2b2c3-a2b3d2 a2b2c2-a3bcd a2b4c-a4bc2 a3bc3-a2b3cd a2bc4-a2b2cd2 a2b2cd-a3bd2 a2b4d-a4bcd a3bc2d-a2b3d2 a2bc3d-a2b2d3 a2bc3-a3c2d a2b3c2-a4c3 a3c4-a2b2c2d a2c5-a2bc2d2 a2bc2d-a3cd2 a2b3cd-a4c2d a3c3d-a2b2cd2 a2c4d-a2bcd3 a2bcd2-a3d3 a2b3d2-a4cd2 a3c2d2-a2b2d3 a2c3d2-a2bd4 ab4c-a2b3d ab6-a3b3c a2b3c2-ab5d ab3c3-ab4d2 ab3c2-a2b2cd ab5c-a3b2c2 a2b2c3-ab4cd ab2c4-ab3cd2 ab3cd-a2b2d2 ab5d-a3b2cd a2b2c2d-ab4d2 ab2c3d-ab3d3 ab2c3-a2bc2d ab4c2-a3bc3 a2bc4-ab3c2d abc5-ab2c2d2 ab2c2d-a2bcd2 ab4cd-a3bc2d a2bc3d-ab3cd2 abc4d-ab2cd3 ab2cd2-a2bd3 ab4d2-a3bcd2 a2bc2d2-ab3d3 abc3d2-ab2d4 abc4-a2c3d ab3c3-a3c4 a2c5-ab2c3d ac6-abc3d2 abc3d-a2c2d2 ab3c2d-a3c3d a2c4d-ab2c2d2 ac5d-abc2d3 abc2d2-a2cd3 ab3cd2-a3c2d2 a2c3d2-ab2cd3 ac4d2-abcd4 abcd3-a2d4 ab3d3-a3cd3 a2c2d3-ab2d4 ac3d3-abd5 b5c-ab4d b7-a2b4c ab4c2-b6d b4c3-b5d2 b4c2-ab3cd b6c-a2b3c2 ab3c3-b5cd b3c4-b4cd2 b4cd-ab3d2 b6d-a2b3cd ab3c2d-b5d2 b3c3d-b4d3 b3c3-ab2c2d b5c2-a2b2c3 ab2c4-b4c2d b2c5-b3c2d2 b3c2d-ab2cd2 b5cd-a2b2c2d ab2c3d-b4cd2 b2c4d-b3cd3 b3cd2-ab2d3 b5d2-a2b2cd2 ab2c2d2-b4d3 b2c3d2-b3d4 b2c4-abc3d b4c3-a2bc4 abc5-b3c3d bc6-b2c3d2 b2c3d-abc2d2 b4c2d-a2bc3d abc4d-b3c2d2 bc5d-b2c2d3 b2c2d2-abcd3 b4cd2-a2bc2d2 abc3d2-b3cd3 bc4d2-b2cd4 b2cd3-abd4 b4d3-a2bcd3 abc2d3-b3d4 bc3d3-b2d5 bc5-ac4d b3c4-a2c5 ac6-b2c4d c7-bc4d2 bc4d-ac3d2 b3c3d-a2c4d ac5d-b2c3d2 c6d-bc3d3 bc3d2-ac2d3 b3c2d2-a2c3d2 ac4d2-b2c2d3 c5d2-bc2d4 bc2d3-acd4 b3cd3-a2c2d3 ac3d3-b2cd4 c4d3-bcd5 bcd4-ad5 b3d4-a2cd4 ac2d4-b2d5 c3d4-bd6 a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 |)|subquotient ({2} | a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 -a3b2 -a4c -a3bd -a3c2 -a2b3 -a3bc -a2b2d -a2bc2 -a2b2c -a3c2 -a2bcd -a2c3 -a2b2d -a3cd -a2bd2 -a2c2d -ab4 -a2b2c -ab3d -ab2c2 -ab3c -a2bc2 -ab2cd -abc3 -ab3d -a2bcd -ab2d2 -abc2d -ab2c2 -a2c3 -abc2d -ac4 -ab2cd -a2c2d -abcd2 -ac3d -ab2d2 -a2cd2 -abd3 -ac2d2 -b5 -ab3c -b4d -b3c2 -b4c -ab2c2 -b3cd -b2c3 -b4d -ab2cd -b3d2 -b2c2d -b3c2 -abc3 -b2c2d -bc4 -b3cd -abc2d -b2cd2 -bc3d -b3d2 -abcd2 -b2d3 -bc2d2 -b2c3 -ac4 -bc3d -c5 -b2c2d -ac3d -bc2d2 -c4d -b2cd2 -ac2d2 -bcd3 -c3d2 -b2d3 -acd3 -bd4 -c2d3 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |, {2} | -a3b2 -a4c -a3bd -a3c2 -a2b3 -a3bc -a2b2d -a2bc2 -a2b2c -a3c2 -a2bcd -a2c3 -a2b2d -a3cd -a2bd2 -a2c2d -ab4 -a2b2c -ab3d -ab2c2 -ab3c -a2bc2 -ab2cd -abc3 -ab3d -a2bcd -ab2d2 -abc2d -ab2c2 -a2c3 -abc2d -ac4 -ab2cd -a2c2d -abcd2 -ac3d -ab2d2 -a2cd2 -abd3 -ac2d2 -b5 -ab3c -b4d -b3c2 -b4c -ab2c2 -b3cd -b2c3 -b4d -ab2cd -b3d2 -b2c2d -b3c2 -abc3 -b2c2d -bc4 -b3cd -abc2d -b2cd2 -bc3d -b3d2 -abcd2 -b2d3 -bc2d2 -b2c3 -ac4 -bc3d -c5 -b2c2d -ac3d -bc2d2 -c4d -b2cd2 -ac2d2 -bcd3 -c3d2 -b2d3 -acd3 -bd4 -c2d3 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |) |subquotient ({4} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 a2d abd acd ad2 b2d bcd bd2 c2d cd2 d3 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |, {4} | a2d abd acd ad2 b2d bcd bd2 c2d cd2 d3 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |) |cokernel {5} | a b c d | | | | | {3} | 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 a3c a3d 0 0 a2bc a2bd 0 0 a2c2 a2cd 0 0 a2cd a2d2 0 0 ab2c ab2d 0 0 abc2 abcd 0 0 abcd abd2 0 0 ac3 ac2d 0 0 ac2d acd2 0 0 acd2 ad3 0 0 b3c b3d 0 0 b2c2 b2cd 0 0 b2cd b2d2 0 0 bc3 bc2d 0 0 bc2d bcd2 0 0 bcd2 bd3 0 0 c4 c3d 0 0 c3d c2d2 0 0 c2d2 cd3 0 0 cd3 d4 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | {3} | a3c a3d 0 0 a2bc a2bd 0 0 a2c2 a2cd 0 0 a2cd a2d2 0 0 ab2c ab2d 0 0 abc2 abcd 0 0 abcd abd2 0 0 ac3 ac2d 0 0 ac2d acd2 0 0 acd2 ad3 0 0 b3c b3d 0 0 b2c2 b2cd 0 0 b2cd b2d2 0 0 bc3 bc2d 0 0 bc2d bcd2 0 0 bcd2 bd3 0 0 c4 c3d 0 0 c3d c2d2 0 0 c2d2 cd3 0 0 cd3 d4 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | | {4} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 -a2c -abc -ac2 -acd -b2c -bc2 -bcd -c3 -c2d -cd2 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | {4} | -a2c -abc -ac2 -acd -b2c -bc2 -bcd -c3 -c2d -cd2 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | | | |{-4, 3} |{-3, 3} | {3} | 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 a4 a3b -a3c -a3d a3b a2b2 -a2bc -a2bd a3c a2bc -a2c2 -a2cd a3d a2bd -a2cd -a2d2 a2b2 ab3 -ab2c -ab2d a2bc ab2c -abc2 -abcd a2bd ab2d -abcd -abd2 a2c2 abc2 -ac3 -ac2d a2cd abcd -ac2d -acd2 a2d2 abd2 -acd2 -ad3 ab3 b4 -b3c -b3d ab2c b3c -b2c2 -b2cd ab2d b3d -b2cd -b2d2 abc2 b2c2 -bc3 -bc2d abcd b2cd -bc2d -bcd2 abd2 b2d2 -bcd2 -bd3 ac3 bc3 -c4 -c3d ac2d bc2d -c3d -c2d2 acd2 bcd2 -c2d2 -cd3 ad3 bd3 -cd3 -d4 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | {3} | a4 a3b -a3c -a3d a3b a2b2 -a2bc -a2bd a3c a2bc -a2c2 -a2cd a3d a2bd -a2cd -a2d2 a2b2 ab3 -ab2c -ab2d a2bc ab2c -abc2 -abcd a2bd ab2d -abcd -abd2 a2c2 abc2 -ac3 -ac2d a2cd abcd -ac2d -acd2 a2d2 abd2 -acd2 -ad3 ab3 b4 -b3c -b3d ab2c b3c -b2c2 -b2cd ab2d b3d -b2cd -b2d2 abc2 b2c2 -bc3 -bc2d abcd b2cd -bc2d -bcd2 abd2 b2d2 -bcd2 -bd3 ac3 bc3 -c4 -c3d ac2d bc2d -c3d -c2d2 acd2 bcd2 -c2d2 -cd3 ad3 bd3 -cd3 -d4 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | | {4} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 -a2b -ab2 -abc -abd -b3 -b2c -b2d -bc2 -bcd -bd2 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | {4} | -a2b -ab2 -abc -abd -b3 -b2c -b2d -bc2 -bcd -bd2 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | |{0, 3} | | | | {3} | 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 a4 a3b 0 0 a3b a2b2 0 0 a3c a2bc 0 0 a3d a2bd 0 0 a2b2 ab3 0 0 a2bc ab2c 0 0 a2bd ab2d 0 0 a2c2 abc2 0 0 a2cd abcd 0 0 a2d2 abd2 0 0 ab3 b4 0 0 ab2c b3c 0 0 ab2d b3d 0 0 abc2 b2c2 0 0 abcd b2cd 0 0 abd2 b2d2 0 0 ac3 bc3 0 0 ac2d bc2d 0 0 acd2 bcd2 0 0 ad3 bd3 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | {3} | 0 0 a4 a3b 0 0 a3b a2b2 0 0 a3c a2bc 0 0 a3d a2bd 0 0 a2b2 ab3 0 0 a2bc ab2c 0 0 a2bd ab2d 0 0 a2c2 abc2 0 0 a2cd abcd 0 0 a2d2 abd2 0 0 ab3 b4 0 0 ab2c b3c 0 0 ab2d b3d 0 0 abc2 b2c2 0 0 abcd b2cd 0 0 abd2 b2d2 0 0 ac3 bc3 0 0 ac2d bc2d 0 0 acd2 bcd2 0 0 ad3 bd3 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | | {4} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | {4} | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | | | | | | | | | | | |{-2, 3} |{-1, 3} | | 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|0 |0 |subquotient (| a2 ab ac ad b2 bc bd c2 cd d2 a3bc-a4d a3b3-a5c a4c2-a3b2d a3c3-a3bd2 a2b2c-a3bd a2b4-a4bc a3bc2-a2b3d a2bc3-a2b2d2 a2bc2-a3cd a2b3c-a4c2 a3c3-a2b2cd a2c4-a2bcd2 a2bcd-a3d2 a2b3d-a4cd a3c2d-a2b2d2 a2c3d-a2bd3 ab3c-a2b2d ab5-a3b2c a2b2c2-ab4d ab2c3-ab3d2 ab2c2-a2bcd ab4c-a3bc2 a2bc3-ab3cd abc4-ab2cd2 ab2cd-a2bd2 ab4d-a3bcd a2bc2d-ab3d2 abc3d-ab2d3 abc3-a2c2d ab3c2-a3c3 a2c4-ab2c2d ac5-abc2d2 abc2d-a2cd2 ab3cd-a3c2d a2c3d-ab2cd2 ac4d-abcd3 abcd2-a2d3 ab3d2-a3cd2 a2c2d2-ab2d3 ac3d2-abd4 b4c-ab3d b6-a2b3c ab3c2-b5d b3c3-b4d2 b3c2-ab2cd b5c-a2b2c2 ab2c3-b4cd b2c4-b3cd2 b3cd-ab2d2 b5d-a2b2cd ab2c2d-b4d2 b2c3d-b3d3 b2c3-abc2d b4c2-a2bc3 abc4-b3c2d bc5-b2c2d2 b2c2d-abcd2 b4cd-a2bc2d abc3d-b3cd2 bc4d-b2cd3 b2cd2-abd3 b4d2-a2bcd2 abc2d2-b3d3 bc3d2-b2d4 bc4-ac3d b3c3-a2c4 ac5-b2c3d c6-bc3d2 bc3d-ac2d2 b3c2d-a2c3d ac4d-b2c2d2 c5d-bc2d3 bc2d2-acd3 b3cd2-a2c2d2 ac3d2-b2cd3 c4d2-bcd4 bcd3-ad4 b3d3-a2cd3 ac2d3-b2d4 c3d3-bd5 a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 |, | a3bc-a4d a3b3-a5c a4c2-a3b2d a3c3-a3bd2 a2b2c-a3bd a2b4-a4bc a3bc2-a2b3d a2bc3-a2b2d2 a2bc2-a3cd a2b3c-a4c2 a3c3-a2b2cd a2c4-a2bcd2 a2bcd-a3d2 a2b3d-a4cd a3c2d-a2b2d2 a2c3d-a2bd3 ab3c-a2b2d ab5-a3b2c a2b2c2-ab4d ab2c3-ab3d2 ab2c2-a2bcd ab4c-a3bc2 a2bc3-ab3cd abc4-ab2cd2 ab2cd-a2bd2 ab4d-a3bcd a2bc2d-ab3d2 abc3d-ab2d3 abc3-a2c2d ab3c2-a3c3 a2c4-ab2c2d ac5-abc2d2 abc2d-a2cd2 ab3cd-a3c2d a2c3d-ab2cd2 ac4d-abcd3 abcd2-a2d3 ab3d2-a3cd2 a2c2d2-ab2d3 ac3d2-abd4 b4c-ab3d b6-a2b3c ab3c2-b5d b3c3-b4d2 b3c2-ab2cd b5c-a2b2c2 ab2c3-b4cd b2c4-b3cd2 b3cd-ab2d2 b5d-a2b2cd ab2c2d-b4d2 b2c3d-b3d3 b2c3-abc2d b4c2-a2bc3 abc4-b3c2d bc5-b2c2d2 b2c2d-abcd2 b4cd-a2bc2d abc3d-b3cd2 bc4d-b2cd3 b2cd2-abd3 b4d2-a2bcd2 abc2d2-b3d3 bc3d2-b2d4 bc4-ac3d b3c3-a2c4 ac5-b2c3d c6-bc3d2 bc3d-ac2d2 b3c2d-a2c3d ac4d-b2c2d2 c5d-bc2d3 bc2d2-acd3 b3cd2-a2c2d2 ac3d2-b2cd3 c4d2-bcd4 bcd3-ad4 b3d3-a2cd3 ac2d3-b2d4 c3d3-bd5 a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 |)|subquotient ({2} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 -a2b2 -a3c -a2bd -a2c2 -ab3 -a2bc -ab2d -abc2 -ab2c -a2c2 -abcd -ac3 -ab2d -a2cd -abd2 -ac2d -b4 -ab2c -b3d -b2c2 -b3c -abc2 -b2cd -bc3 -b3d -abcd -b2d2 -bc2d -b2c2 -ac3 -bc2d -c4 -b2cd -ac2d -bcd2 -c3d -b2d2 -acd2 -bd3 -c2d2 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |, {2} | -a2b2 -a3c -a2bd -a2c2 -ab3 -a2bc -ab2d -abc2 -ab2c -a2c2 -abcd -ac3 -ab2d -a2cd -abd2 -ac2d -b4 -ab2c -b3d -b2c2 -b3c -abc2 -b2cd -bc3 -b3d -abcd -b2d2 -bc2d -b2c2 -ac3 -bc2d -c4 -b2cd -ac2d -bcd2 -c3d -b2d2 -acd2 -bd3 -c2d2 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |) |cokernel {4} | ad bd cd d2 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 | | | | | | {3} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 a2c a2d 0 0 abc abd 0 0 ac2 acd 0 0 acd ad2 0 0 b2c b2d 0 0 bc2 bcd 0 0 bcd bd2 0 0 c3 c2d 0 0 c2d cd2 0 0 cd2 d3 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | {3} | a2c a2d 0 0 abc abd 0 0 ac2 acd 0 0 acd ad2 0 0 b2c b2d 0 0 bc2 bcd 0 0 bcd bd2 0 0 c3 c2d 0 0 c2d cd2 0 0 cd2 d3 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | | {4} | -ac -bc -c2 -cd 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | | |{-4, 2} |{-3, 2} |{-2, 2} | {3} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 a3 a2b -a2c -a2d a2b ab2 -abc -abd a2c abc -ac2 -acd a2d abd -acd -ad2 ab2 b3 -b2c -b2d abc b2c -bc2 -bcd abd b2d -bcd -bd2 ac2 bc2 -c3 -c2d acd bcd -c2d -cd2 ad2 bd2 -cd2 -d3 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | {3} | a3 a2b -a2c -a2d a2b ab2 -abc -abd a2c abc -ac2 -acd a2d abd -acd -ad2 ab2 b3 -b2c -b2d abc b2c -bc2 -bcd abd b2d -bcd -bd2 ac2 bc2 -c3 -c2d acd bcd -c2d -cd2 ad2 bd2 -cd2 -d3 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | | {4} | -ab -b2 -bc -bd 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | | | | | | {3} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 a3 a2b 0 0 a2b ab2 0 0 a2c abc 0 0 a2d abd 0 0 ab2 b3 0 0 abc b2c 0 0 abd b2d 0 0 ac2 bc2 0 0 acd bcd 0 0 ad2 bd2 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | {3} | 0 0 a3 a2b 0 0 a2b ab2 0 0 a2c abc 0 0 a2d abd 0 0 ab2 b3 0 0 abc b2c 0 0 abd b2d 0 0 ac2 bc2 0 0 acd bcd 0 0 ad2 bd2 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | | {4} | a2 ab ac ad 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | | | | | | | | | | | |{-1, 2} |{0, 2} | +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------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|0 |0 |0 |subquotient (| a b c d a2bc-a3d a2b3-a4c a3c2-a2b2d a2c3-a2bd2 ab2c-a2bd ab4-a3bc a2bc2-ab3d abc3-ab2d2 abc2-a2cd ab3c-a3c2 a2c3-ab2cd ac4-abcd2 abcd-a2d2 ab3d-a3cd a2c2d-ab2d2 ac3d-abd3 b3c-ab2d b5-a2b2c ab2c2-b4d b2c3-b3d2 b2c2-abcd b4c-a2bc2 abc3-b3cd bc4-b2cd2 b2cd-abd2 b4d-a2bcd abc2d-b3d2 bc3d-b2d3 bc3-ac2d b3c2-a2c3 ac4-b2c2d c5-bc2d2 bc2d-acd2 b3cd-a2c2d ac3d-b2cd2 c4d-bcd3 bcd2-ad3 b3d2-a2cd2 ac2d2-b2d3 c3d2-bd4 a2 ab ac ad b2 bc bd c2 cd d2 |, | a2bc-a3d a2b3-a4c a3c2-a2b2d a2c3-a2bd2 ab2c-a2bd ab4-a3bc a2bc2-ab3d abc3-ab2d2 abc2-a2cd ab3c-a3c2 a2c3-ab2cd ac4-abcd2 abcd-a2d2 ab3d-a3cd a2c2d-ab2d2 ac3d-abd3 b3c-ab2d b5-a2b2c ab2c2-b4d b2c3-b3d2 b2c2-abcd b4c-a2bc2 abc3-b3cd bc4-b2cd2 b2cd-abd2 b4d-a2bcd abc2d-b3d2 bc3d-b2d3 bc3-ac2d b3c2-a2c3 ac4-b2c2d c5-bc2d2 bc2d-acd2 b3cd-a2c2d ac3d-b2cd2 c4d-bcd3 bcd2-ad3 b3d2-a2cd2 ac2d2-b2d3 c3d2-bd4 a2 ab ac ad b2 bc bd c2 cd d2 |)|cokernel {2} | -ab2 -a2c -abd -ac2 -b3 -abc -b2d -bc2 -b2c -ac2 -bcd -c3 -b2d -acd -bd2 -c2d a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 | | | | | | | {3} | ac ad 0 0 bc bd 0 0 c2 cd 0 0 cd d2 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | | |{-4, 1} |{-3, 1} |{-2, 1} |{-1, 1} | {3} | a2 ab -ac -ad ab b2 -bc -bd ac bc -c2 -cd ad bd -cd -d2 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | | | | | | | {3} | 0 0 a2 ab 0 0 ab b2 0 0 ac bc 0 0 ad bd 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | | | | | | | | | | | | |{0, 1} | 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|0 |0 |0 |0 |cokernel | abc-a2d ab3-a3c a2c2-ab2d ac3-abd2 b2c-abd b4-a2bc abc2-b3d bc3-b2d2 bc2-acd b3c-a2c2 ac3-b2cd c4-bcd2 bcd-ad2 b3d-a2cd ac2d-b2d2 c3d-bd3 a b c d || | | | | | | |{-4, 0} |{-3, 0} |{-2, 0} |{-1, 0} |{0, 0} | 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o6 : SpectralSequencePage``` ```i7 : E^1 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o7 = |subquotient (| a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 a4bc-a5d a4b3-a6c a5c2-a4b2d a4c3-a4bd2 a3b2c-a4bd a3b4-a5bc a4bc2-a3b3d a3bc3-a3b2d2 a3bc2-a4cd a3b3c-a5c2 a4c3-a3b2cd a3c4-a3bcd2 a3bcd-a4d2 a3b3d-a5cd a4c2d-a3b2d2 a3c3d-a3bd3 a2b3c-a3b2d a2b5-a4b2c a3b2c2-a2b4d a2b2c3-a2b3d2 a2b2c2-a3bcd a2b4c-a4bc2 a3bc3-a2b3cd a2bc4-a2b2cd2 a2b2cd-a3bd2 a2b4d-a4bcd a3bc2d-a2b3d2 a2bc3d-a2b2d3 a2bc3-a3c2d a2b3c2-a4c3 a3c4-a2b2c2d a2c5-a2bc2d2 a2bc2d-a3cd2 a2b3cd-a4c2d a3c3d-a2b2cd2 a2c4d-a2bcd3 a2bcd2-a3d3 a2b3d2-a4cd2 a3c2d2-a2b2d3 a2c3d2-a2bd4 ab4c-a2b3d ab6-a3b3c a2b3c2-ab5d ab3c3-ab4d2 ab3c2-a2b2cd ab5c-a3b2c2 a2b2c3-ab4cd ab2c4-ab3cd2 ab3cd-a2b2d2 ab5d-a3b2cd a2b2c2d-ab4d2 ab2c3d-ab3d3 ab2c3-a2bc2d ab4c2-a3bc3 a2bc4-ab3c2d abc5-ab2c2d2 ab2c2d-a2bcd2 ab4cd-a3bc2d a2bc3d-ab3cd2 abc4d-ab2cd3 ab2cd2-a2bd3 ab4d2-a3bcd2 a2bc2d2-ab3d3 abc3d2-ab2d4 abc4-a2c3d ab3c3-a3c4 a2c5-ab2c3d ac6-abc3d2 abc3d-a2c2d2 ab3c2d-a3c3d a2c4d-ab2c2d2 ac5d-abc2d3 abc2d2-a2cd3 ab3cd2-a3c2d2 a2c3d2-ab2cd3 ac4d2-abcd4 abcd3-a2d4 ab3d3-a3cd3 a2c2d3-ab2d4 ac3d3-abd5 b5c-ab4d b7-a2b4c ab4c2-b6d b4c3-b5d2 b4c2-ab3cd b6c-a2b3c2 ab3c3-b5cd b3c4-b4cd2 b4cd-ab3d2 b6d-a2b3cd ab3c2d-b5d2 b3c3d-b4d3 b3c3-ab2c2d b5c2-a2b2c3 ab2c4-b4c2d b2c5-b3c2d2 b3c2d-ab2cd2 b5cd-a2b2c2d ab2c3d-b4cd2 b2c4d-b3cd3 b3cd2-ab2d3 b5d2-a2b2cd2 ab2c2d2-b4d3 b2c3d2-b3d4 b2c4-abc3d b4c3-a2bc4 abc5-b3c3d bc6-b2c3d2 b2c3d-abc2d2 b4c2d-a2bc3d abc4d-b3c2d2 bc5d-b2c2d3 b2c2d2-abcd3 b4cd2-a2bc2d2 abc3d2-b3cd3 bc4d2-b2cd4 b2cd3-abd4 b4d3-a2bcd3 abc2d3-b3d4 bc3d3-b2d5 bc5-ac4d b3c4-a2c5 ac6-b2c4d c7-bc4d2 bc4d-ac3d2 b3c3d-a2c4d ac5d-b2c3d2 c6d-bc3d3 bc3d2-ac2d3 b3c2d2-a2c3d2 ac4d2-b2c2d3 c5d2-bc2d4 bc2d3-acd4 b3cd3-a2c2d3 ac3d3-b2cd4 c4d3-bcd5 bcd4-ad5 b3d4-a2cd4 ac2d4-b2d5 c3d4-bd6 0 |, | a4bc-a5d a4b3-a6c a5c2-a4b2d a4c3-a4bd2 a3b2c-a4bd a3b4-a5bc a4bc2-a3b3d a3bc3-a3b2d2 a3bc2-a4cd a3b3c-a5c2 a4c3-a3b2cd a3c4-a3bcd2 a3bcd-a4d2 a3b3d-a5cd a4c2d-a3b2d2 a3c3d-a3bd3 a2b3c-a3b2d a2b5-a4b2c a3b2c2-a2b4d a2b2c3-a2b3d2 a2b2c2-a3bcd a2b4c-a4bc2 a3bc3-a2b3cd a2bc4-a2b2cd2 a2b2cd-a3bd2 a2b4d-a4bcd a3bc2d-a2b3d2 a2bc3d-a2b2d3 a2bc3-a3c2d a2b3c2-a4c3 a3c4-a2b2c2d a2c5-a2bc2d2 a2bc2d-a3cd2 a2b3cd-a4c2d a3c3d-a2b2cd2 a2c4d-a2bcd3 a2bcd2-a3d3 a2b3d2-a4cd2 a3c2d2-a2b2d3 a2c3d2-a2bd4 ab4c-a2b3d ab6-a3b3c a2b3c2-ab5d ab3c3-ab4d2 ab3c2-a2b2cd ab5c-a3b2c2 a2b2c3-ab4cd ab2c4-ab3cd2 ab3cd-a2b2d2 ab5d-a3b2cd a2b2c2d-ab4d2 ab2c3d-ab3d3 ab2c3-a2bc2d ab4c2-a3bc3 a2bc4-ab3c2d abc5-ab2c2d2 ab2c2d-a2bcd2 ab4cd-a3bc2d a2bc3d-ab3cd2 abc4d-ab2cd3 ab2cd2-a2bd3 ab4d2-a3bcd2 a2bc2d2-ab3d3 abc3d2-ab2d4 abc4-a2c3d ab3c3-a3c4 a2c5-ab2c3d ac6-abc3d2 abc3d-a2c2d2 ab3c2d-a3c3d a2c4d-ab2c2d2 ac5d-abc2d3 abc2d2-a2cd3 ab3cd2-a3c2d2 a2c3d2-ab2cd3 ac4d2-abcd4 abcd3-a2d4 ab3d3-a3cd3 a2c2d3-ab2d4 ac3d3-abd5 b5c-ab4d b7-a2b4c ab4c2-b6d b4c3-b5d2 b4c2-ab3cd b6c-a2b3c2 ab3c3-b5cd b3c4-b4cd2 b4cd-ab3d2 b6d-a2b3cd ab3c2d-b5d2 b3c3d-b4d3 b3c3-ab2c2d b5c2-a2b2c3 ab2c4-b4c2d b2c5-b3c2d2 b3c2d-ab2cd2 b5cd-a2b2c2d ab2c3d-b4cd2 b2c4d-b3cd3 b3cd2-ab2d3 b5d2-a2b2cd2 ab2c2d2-b4d3 b2c3d2-b3d4 b2c4-abc3d b4c3-a2bc4 abc5-b3c3d bc6-b2c3d2 b2c3d-abc2d2 b4c2d-a2bc3d abc4d-b3c2d2 bc5d-b2c2d3 b2c2d2-abcd3 b4cd2-a2bc2d2 abc3d2-b3cd3 bc4d2-b2cd4 b2cd3-abd4 b4d3-a2bcd3 abc2d3-b3d4 bc3d3-b2d5 bc5-ac4d b3c4-a2c5 ac6-b2c4d c7-bc4d2 bc4d-ac3d2 b3c3d-a2c4d ac5d-b2c3d2 c6d-bc3d3 bc3d2-ac2d3 b3c2d2-a2c3d2 ac4d2-b2c2d3 c5d2-bc2d4 bc2d3-acd4 b3cd3-a2c2d3 ac3d3-b2cd4 c4d3-bcd5 bcd4-ad5 b3d4-a2cd4 ac2d4-b2d5 c3d4-bd6 0 |)|subquotient ({2} | a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 -a3b2 -a4c -a3bd -a3c2 -a2b3 -a3bc -a2b2d -a2bc2 -a2b2c -a3c2 -a2bcd -a2c3 -a2b2d -a3cd -a2bd2 -a2c2d -ab4 -a2b2c -ab3d -ab2c2 -ab3c -a2bc2 -ab2cd -abc3 -ab3d -a2bcd -ab2d2 -abc2d -ab2c2 -a2c3 -abc2d -ac4 -ab2cd -a2c2d -abcd2 -ac3d -ab2d2 -a2cd2 -abd3 -ac2d2 -b5 -ab3c -b4d -b3c2 -b4c -ab2c2 -b3cd -b2c3 -b4d -ab2cd -b3d2 -b2c2d -b3c2 -abc3 -b2c2d -bc4 -b3cd -abc2d -b2cd2 -bc3d -b3d2 -abcd2 -b2d3 -bc2d2 -b2c3 -ac4 -bc3d -c5 -b2c2d -ac3d -bc2d2 -c4d -b2cd2 -ac2d2 -bcd3 -c3d2 -b2d3 -acd3 -bd4 -c2d3 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 |, {2} | -a3b2 -a4c -a3bd -a3c2 -a2b3 -a3bc -a2b2d -a2bc2 -a2b2c -a3c2 -a2bcd -a2c3 -a2b2d -a3cd -a2bd2 -a2c2d -ab4 -a2b2c -ab3d -ab2c2 -ab3c -a2bc2 -ab2cd -abc3 -ab3d -a2bcd -ab2d2 -abc2d -ab2c2 -a2c3 -abc2d -ac4 -ab2cd -a2c2d -abcd2 -ac3d -ab2d2 -a2cd2 -abd3 -ac2d2 -b5 -ab3c -b4d -b3c2 -b4c -ab2c2 -b3cd -b2c3 -b4d -ab2cd -b3d2 -b2c2d -b3c2 -abc3 -b2c2d -bc4 -b3cd -abc2d -b2cd2 -bc3d -b3d2 -abcd2 -b2d3 -bc2d2 -b2c3 -ac4 -bc3d -c5 -b2c2d -ac3d -bc2d2 -c4d -b2cd2 -ac2d2 -bcd3 -c3d2 -b2d3 -acd3 -bd4 -c2d3 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 |)|subquotient ({4} | a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 a2d abd acd ad2 b2d bcd bd2 c2d cd2 d3 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |, {4} | a2d abd acd ad2 b2d bcd bd2 c2d cd2 d3 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |) |subquotient ({5} | a b c d a2 ab ac ad b2 bc bd c2 cd d2 |, {5} | a2 ab ac ad b2 bc bd c2 cd d2 |) |0 | | | {3} | 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 a3c a3d 0 0 a2bc a2bd 0 0 a2c2 a2cd 0 0 a2cd a2d2 0 0 ab2c ab2d 0 0 abc2 abcd 0 0 abcd abd2 0 0 ac3 ac2d 0 0 ac2d acd2 0 0 acd2 ad3 0 0 b3c b3d 0 0 b2c2 b2cd 0 0 b2cd b2d2 0 0 bc3 bc2d 0 0 bc2d bcd2 0 0 bcd2 bd3 0 0 c4 c3d 0 0 c3d c2d2 0 0 c2d2 cd3 0 0 cd3 d4 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 | {3} | a3c a3d 0 0 a2bc a2bd 0 0 a2c2 a2cd 0 0 a2cd a2d2 0 0 ab2c ab2d 0 0 abc2 abcd 0 0 abcd abd2 0 0 ac3 ac2d 0 0 ac2d acd2 0 0 acd2 ad3 0 0 b3c b3d 0 0 b2c2 b2cd 0 0 b2cd b2d2 0 0 bc3 bc2d 0 0 bc2d bcd2 0 0 bcd2 bd3 0 0 c4 c3d 0 0 c3d c2d2 0 0 c2d2 cd3 0 0 cd3 d4 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 | | {4} | 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 -a2c -abc -ac2 -acd -b2c -bc2 -bcd -c3 -c2d -cd2 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | {4} | -a2c -abc -ac2 -acd -b2c -bc2 -bcd -c3 -c2d -cd2 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | | | | |{-4, 4} | {3} | 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 a4 a3b -a3c -a3d a3b a2b2 -a2bc -a2bd a3c a2bc -a2c2 -a2cd a3d a2bd -a2cd -a2d2 a2b2 ab3 -ab2c -ab2d a2bc ab2c -abc2 -abcd a2bd ab2d -abcd -abd2 a2c2 abc2 -ac3 -ac2d a2cd abcd -ac2d -acd2 a2d2 abd2 -acd2 -ad3 ab3 b4 -b3c -b3d ab2c b3c -b2c2 -b2cd ab2d b3d -b2cd -b2d2 abc2 b2c2 -bc3 -bc2d abcd b2cd -bc2d -bcd2 abd2 b2d2 -bcd2 -bd3 ac3 bc3 -c4 -c3d ac2d bc2d -c3d -c2d2 acd2 bcd2 -c2d2 -cd3 ad3 bd3 -cd3 -d4 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | {3} | a4 a3b -a3c -a3d a3b a2b2 -a2bc -a2bd a3c a2bc -a2c2 -a2cd a3d a2bd -a2cd -a2d2 a2b2 ab3 -ab2c -ab2d a2bc ab2c -abc2 -abcd a2bd ab2d -abcd -abd2 a2c2 abc2 -ac3 -ac2d a2cd abcd -ac2d -acd2 a2d2 abd2 -acd2 -ad3 ab3 b4 -b3c -b3d ab2c b3c -b2c2 -b2cd ab2d b3d -b2cd -b2d2 abc2 b2c2 -bc3 -bc2d abcd b2cd -bc2d -bcd2 abd2 b2d2 -bcd2 -bd3 ac3 bc3 -c4 -c3d ac2d bc2d -c3d -c2d2 acd2 bcd2 -c2d2 -cd3 ad3 bd3 -cd3 -d4 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | | {4} | 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 -a2b -ab2 -abc -abd -b3 -b2c -b2d -bc2 -bcd -bd2 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | {4} | -a2b -ab2 -abc -abd -b3 -b2c -b2d -bc2 -bcd -bd2 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | |{-1, 4} |{0, 4} | | | {3} | 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 a4 a3b 0 0 a3b a2b2 0 0 a3c a2bc 0 0 a3d a2bd 0 0 a2b2 ab3 0 0 a2bc ab2c 0 0 a2bd ab2d 0 0 a2c2 abc2 0 0 a2cd abcd 0 0 a2d2 abd2 0 0 ab3 b4 0 0 ab2c b3c 0 0 ab2d b3d 0 0 abc2 b2c2 0 0 abcd b2cd 0 0 abd2 b2d2 0 0 ac3 bc3 0 0 ac2d bc2d 0 0 acd2 bcd2 0 0 ad3 bd3 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 | {3} | 0 0 a4 a3b 0 0 a3b a2b2 0 0 a3c a2bc 0 0 a3d a2bd 0 0 a2b2 ab3 0 0 a2bc ab2c 0 0 a2bd ab2d 0 0 a2c2 abc2 0 0 a2cd abcd 0 0 a2d2 abd2 0 0 ab3 b4 0 0 ab2c b3c 0 0 ab2d b3d 0 0 abc2 b2c2 0 0 abcd b2cd 0 0 abd2 b2d2 0 0 ac3 bc3 0 0 ac2d bc2d 0 0 acd2 bcd2 0 0 ad3 bd3 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 | | {4} | 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | {4} | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | | | | | | | | | | | |{-3, 4} |{-2, 4} | | | +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------+ |0 |subquotient (| a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 a3bc-a4d a3b3-a5c a4c2-a3b2d a3c3-a3bd2 a2b2c-a3bd a2b4-a4bc a3bc2-a2b3d a2bc3-a2b2d2 a2bc2-a3cd a2b3c-a4c2 a3c3-a2b2cd a2c4-a2bcd2 a2bcd-a3d2 a2b3d-a4cd a3c2d-a2b2d2 a2c3d-a2bd3 ab3c-a2b2d ab5-a3b2c a2b2c2-ab4d ab2c3-ab3d2 ab2c2-a2bcd ab4c-a3bc2 a2bc3-ab3cd abc4-ab2cd2 ab2cd-a2bd2 ab4d-a3bcd a2bc2d-ab3d2 abc3d-ab2d3 abc3-a2c2d ab3c2-a3c3 a2c4-ab2c2d ac5-abc2d2 abc2d-a2cd2 ab3cd-a3c2d a2c3d-ab2cd2 ac4d-abcd3 abcd2-a2d3 ab3d2-a3cd2 a2c2d2-ab2d3 ac3d2-abd4 b4c-ab3d b6-a2b3c ab3c2-b5d b3c3-b4d2 b3c2-ab2cd b5c-a2b2c2 ab2c3-b4cd b2c4-b3cd2 b3cd-ab2d2 b5d-a2b2cd ab2c2d-b4d2 b2c3d-b3d3 b2c3-abc2d b4c2-a2bc3 abc4-b3c2d bc5-b2c2d2 b2c2d-abcd2 b4cd-a2bc2d abc3d-b3cd2 bc4d-b2cd3 b2cd2-abd3 b4d2-a2bcd2 abc2d2-b3d3 bc3d2-b2d4 bc4-ac3d b3c3-a2c4 ac5-b2c3d c6-bc3d2 bc3d-ac2d2 b3c2d-a2c3d ac4d-b2c2d2 c5d-bc2d3 bc2d2-acd3 b3cd2-a2c2d2 ac3d2-b2cd3 c4d2-bcd4 bcd3-ad4 b3d3-a2cd3 ac2d3-b2d4 c3d3-bd5 a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 |, | a3bc-a4d a3b3-a5c a4c2-a3b2d a3c3-a3bd2 a2b2c-a3bd a2b4-a4bc a3bc2-a2b3d a2bc3-a2b2d2 a2bc2-a3cd a2b3c-a4c2 a3c3-a2b2cd a2c4-a2bcd2 a2bcd-a3d2 a2b3d-a4cd a3c2d-a2b2d2 a2c3d-a2bd3 ab3c-a2b2d ab5-a3b2c a2b2c2-ab4d ab2c3-ab3d2 ab2c2-a2bcd ab4c-a3bc2 a2bc3-ab3cd abc4-ab2cd2 ab2cd-a2bd2 ab4d-a3bcd a2bc2d-ab3d2 abc3d-ab2d3 abc3-a2c2d ab3c2-a3c3 a2c4-ab2c2d ac5-abc2d2 abc2d-a2cd2 ab3cd-a3c2d a2c3d-ab2cd2 ac4d-abcd3 abcd2-a2d3 ab3d2-a3cd2 a2c2d2-ab2d3 ac3d2-abd4 b4c-ab3d b6-a2b3c ab3c2-b5d b3c3-b4d2 b3c2-ab2cd b5c-a2b2c2 ab2c3-b4cd b2c4-b3cd2 b3cd-ab2d2 b5d-a2b2cd ab2c2d-b4d2 b2c3d-b3d3 b2c3-abc2d b4c2-a2bc3 abc4-b3c2d bc5-b2c2d2 b2c2d-abcd2 b4cd-a2bc2d abc3d-b3cd2 bc4d-b2cd3 b2cd2-abd3 b4d2-a2bcd2 abc2d2-b3d3 bc3d2-b2d4 bc4-ac3d b3c3-a2c4 ac5-b2c3d c6-bc3d2 bc3d-ac2d2 b3c2d-a2c3d ac4d-b2c2d2 c5d-bc2d3 bc2d2-acd3 b3cd2-a2c2d2 ac3d2-b2cd3 c4d2-bcd4 bcd3-ad4 b3d3-a2cd3 ac2d3-b2d4 c3d3-bd5 a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 |) |subquotient ({2} | a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 -a2b2 -a3c -a2bd -a2c2 -ab3 -a2bc -ab2d -abc2 -ab2c -a2c2 -abcd -ac3 -ab2d -a2cd -abd2 -ac2d -b4 -ab2c -b3d -b2c2 -b3c -abc2 -b2cd -bc3 -b3d -abcd -b2d2 -bc2d -b2c2 -ac3 -bc2d -c4 -b2cd -ac2d -bcd2 -c3d -b2d2 -acd2 -bd3 -c2d2 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |, {2} | -a2b2 -a3c -a2bd -a2c2 -ab3 -a2bc -ab2d -abc2 -ab2c -a2c2 -abcd -ac3 -ab2d -a2cd -abd2 -ac2d -b4 -ab2c -b3d -b2c2 -b3c -abc2 -b2cd -bc3 -b3d -abcd -b2d2 -bc2d -b2c2 -ac3 -bc2d -c4 -b2cd -ac2d -bcd2 -c3d -b2d2 -acd2 -bd3 -c2d2 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |)|subquotient ({4} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 ad bd cd d2 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |, {4} | ad bd cd d2 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |) |cokernel {5} | a b c d | | | | | {3} | 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 a2c a2d 0 0 abc abd 0 0 ac2 acd 0 0 acd ad2 0 0 b2c b2d 0 0 bc2 bcd 0 0 bcd bd2 0 0 c3 c2d 0 0 c2d cd2 0 0 cd2 d3 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | {3} | a2c a2d 0 0 abc abd 0 0 ac2 acd 0 0 acd ad2 0 0 b2c b2d 0 0 bc2 bcd 0 0 bcd bd2 0 0 c3 c2d 0 0 c2d cd2 0 0 cd2 d3 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | | {4} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 -ac -bc -c2 -cd 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | {4} | -ac -bc -c2 -cd 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | | | |{-4, 3} |{-3, 3} | {3} | 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 a3 a2b -a2c -a2d a2b ab2 -abc -abd a2c abc -ac2 -acd a2d abd -acd -ad2 ab2 b3 -b2c -b2d abc b2c -bc2 -bcd abd b2d -bcd -bd2 ac2 bc2 -c3 -c2d acd bcd -c2d -cd2 ad2 bd2 -cd2 -d3 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | {3} | a3 a2b -a2c -a2d a2b ab2 -abc -abd a2c abc -ac2 -acd a2d abd -acd -ad2 ab2 b3 -b2c -b2d abc b2c -bc2 -bcd abd b2d -bcd -bd2 ac2 bc2 -c3 -c2d acd bcd -c2d -cd2 ad2 bd2 -cd2 -d3 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | | {4} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 -ab -b2 -bc -bd 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | {4} | -ab -b2 -bc -bd 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | |{0, 3} | | | | {3} | 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 a3 a2b 0 0 a2b ab2 0 0 a2c abc 0 0 a2d abd 0 0 ab2 b3 0 0 abc b2c 0 0 abd b2d 0 0 ac2 bc2 0 0 acd bcd 0 0 ad2 bd2 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | {3} | 0 0 a3 a2b 0 0 a2b ab2 0 0 a2c abc 0 0 a2d abd 0 0 ab2 b3 0 0 abc b2c 0 0 abd b2d 0 0 ac2 bc2 0 0 acd bcd 0 0 ad2 bd2 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | | {4} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d a2 ab ac ad 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | {4} | a2 ab ac ad 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | | | | | | | | | | | |{-2, 3} |{-1, 3} | | 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|0 |0 |subquotient (| a2 ab ac ad b2 bc bd c2 cd d2 a2bc-a3d a2b3-a4c a3c2-a2b2d a2c3-a2bd2 ab2c-a2bd ab4-a3bc a2bc2-ab3d abc3-ab2d2 abc2-a2cd ab3c-a3c2 a2c3-ab2cd ac4-abcd2 abcd-a2d2 ab3d-a3cd a2c2d-ab2d2 ac3d-abd3 b3c-ab2d b5-a2b2c ab2c2-b4d b2c3-b3d2 b2c2-abcd b4c-a2bc2 abc3-b3cd bc4-b2cd2 b2cd-abd2 b4d-a2bcd abc2d-b3d2 bc3d-b2d3 bc3-ac2d b3c2-a2c3 ac4-b2c2d c5-bc2d2 bc2d-acd2 b3cd-a2c2d ac3d-b2cd2 c4d-bcd3 bcd2-ad3 b3d2-a2cd2 ac2d2-b2d3 c3d2-bd4 a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 |, | a2bc-a3d a2b3-a4c a3c2-a2b2d a2c3-a2bd2 ab2c-a2bd ab4-a3bc a2bc2-ab3d abc3-ab2d2 abc2-a2cd ab3c-a3c2 a2c3-ab2cd ac4-abcd2 abcd-a2d2 ab3d-a3cd a2c2d-ab2d2 ac3d-abd3 b3c-ab2d b5-a2b2c ab2c2-b4d b2c3-b3d2 b2c2-abcd b4c-a2bc2 abc3-b3cd bc4-b2cd2 b2cd-abd2 b4d-a2bcd abc2d-b3d2 bc3d-b2d3 bc3-ac2d b3c2-a2c3 ac4-b2c2d c5-bc2d2 bc2d-acd2 b3cd-a2c2d ac3d-b2cd2 c4d-bcd3 bcd2-ad3 b3d2-a2cd2 ac2d2-b2d3 c3d2-bd4 a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 |) |subquotient ({2} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 -ab2 -a2c -abd -ac2 -b3 -abc -b2d -bc2 -b2c -ac2 -bcd -c3 -b2d -acd -bd2 -c2d a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |, {2} | -ab2 -a2c -abd -ac2 -b3 -abc -b2d -bc2 -b2c -ac2 -bcd -c3 -b2d -acd -bd2 -c2d a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |)|cokernel {4} | d a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 | | | | | | {3} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 ac ad 0 0 bc bd 0 0 c2 cd 0 0 cd d2 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | {3} | ac ad 0 0 bc bd 0 0 c2 cd 0 0 cd d2 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | | {4} | -c 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | | |{-4, 2} |{-3, 2} |{-2, 2} | {3} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 a2 ab -ac -ad ab b2 -bc -bd ac bc -c2 -cd ad bd -cd -d2 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | {3} | a2 ab -ac -ad ab b2 -bc -bd ac bc -c2 -cd ad bd -cd -d2 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | | {4} | -b 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | | | | | | {3} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 a2 ab 0 0 ab b2 0 0 ac bc 0 0 ad bd 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | {3} | 0 0 a2 ab 0 0 ab b2 0 0 ac bc 0 0 ad bd 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | | {4} | a 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | | | | | | | | | | | |{-1, 2} |{0, 2} | 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|0 |0 |0 |subquotient (| a b c d abc-a2d ab3-a3c a2c2-ab2d ac3-abd2 b2c-abd b4-a2bc abc2-b3d bc3-b2d2 bc2-acd b3c-a2c2 ac3-b2cd c4-bcd2 bcd-ad2 b3d-a2cd ac2d-b2d2 c3d-bd3 a2 ab ac ad b2 bc bd c2 cd d2 |, | abc-a2d ab3-a3c a2c2-ab2d ac3-abd2 b2c-abd b4-a2bc abc2-b3d bc3-b2d2 bc2-acd b3c-a2c2 ac3-b2cd c4-bcd2 bcd-ad2 b3d-a2cd ac2d-b2d2 c3d-bd3 a2 ab ac ad b2 bc bd c2 cd d2 |) |cokernel {2} | -b2 -ac -bd -c2 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 || | | | | | {3} | c d 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 || |{-4, 1} |{-3, 1} |{-2, 1} |{-1, 1} | {3} | a b -c -d 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 || | | | | | {3} | 0 0 a b 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d || | | | | | | | | | | |{0, 1} | 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|0 |0 |0 |0 |cokernel | bc-ad b3-a2c ac2-b2d c3-bd2 a b c d | | | | | | | | |{-4, 0} |{-3, 0} |{-2, 0} |{-1, 0} |{0, 0} | 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o7 : SpectralSequencePage``` ```i8 : E^infinity 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o8 = |subquotient (| a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 bc-ad b3-a2c ac2-b2d c3-bd2 0 |, | bc-ad b3-a2c ac2-b2d c3-bd2 0 |)|subquotient ({2} | 0 a2b2 a2bd a2bd 0 a3c a3c a2c2 a2b2 ab3 ab2d ab2d a2bd a2bc a2bc a2bc abc2 abc2 a2c2 ab2c ab2c abcd abcd abcd 0 abd2 ac3 abd2 0 a2cd a2cd a2cd ac2d ac2d ab3 ab2c b4 b3d b3d ab2d 0 b2c2 b2c2 abc2 b3c b3c b2cd b2cd b2cd abcd 0 b2d2 bc3 b2d2 abd2 0 bc2d bc2d bc3 ac3 0 0 bc2d ac2d 0 bcd2 bcd2 c4 bcd2 0 bd3 c3d c3d 0 bd3 0 acd2 acd2 acd2 c2d2 c2d2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b2 -ac -bd -c2 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 |, {2} | -b2 -ac -bd -c2 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 |)|subquotient ({4} | ad bd cd d2 a2d 0 0 -abd 0 0 0 0 -acd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b2d -abd 0 0 0 -bcd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c2d -bcd -acd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |, {4} | d a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |) |subquotient ({5} | 0 0 0 0 0 0 a2 ab ac ad b2 bc bd c2 cd d2 |, {5} | a2 ab ac ad b2 bc bd c2 cd d2 |) |0 | | | {3} | 0 -a2c 0 0 0 -a2d -a2d 0 -a2c -abc 0 0 0 -abd -abd -abd 0 0 0 -ac2 -ac2 0 0 0 acd 0 0 0 0 -ad2 -ad2 -ad2 0 0 -abc -ac2 -b2c 0 0 0 -ac2+b2d 0 0 0 -bc2 -bc2 0 0 0 0 bcd 0 0 0 0 bd2 0 0 0 0 c3 0 0 0 c2d 0 0 0 0 cd2 0 0 0 0 0 0 -d3 -d3 -d3 0 0 0 0 -a2cd -a2cd 0 0 0 0 0 0 0 0 0 0 0 0 0 -a2c2+ab2d 0 -abcd -abcd 0 0 0 0 0 0 -ac3+abd2 0 0 0 0 0 0 0 0 0 -abc2+b3d -a2c2+ab2d -ac3+abd2 -ac3+b2cd+abd2 0 0 0 0 0 -bc3+ac2d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c d 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 | {3} | c d 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 | | {4} | -ac -bc -c2 -cd -a2c 0 0 abc 0 0 0 0 ac2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2c abc 0 0 0 bc2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c3 bc2 ac2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | {4} | -c 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | | | | |{-4, 4} | {3} | 0 -a3 a2c a2c 0 -a2b -a2b a2d -a3 -a2b abc abc a2c -ab2 -ab2 -ab2 abd abd a2d -a2c -a2c ac2 ac2 ac2 abc+a2d acd acd acd 0 -abd -abd -abd ad2 ad2 -a2b -a2c -ab2 b2c b2c abc b3-a2c b2d b2d abd -abc -abc bc2 bc2 bc2 ac2 b2c+abd bcd bcd bcd acd ac2+b2d bd2 bd2 bcd acd ac2+b2d c3-bd2 bd2 ad2 bc2+acd c2d c2d c2d c2d bcd+ad2 cd2 cd2 cd2 0 cd2 0 -bd2 -bd2 -bd2 d3 d3 0 0 -a2bc-a3d -a2bc-a3d 0 0 0 0 0 0 0 0 0 0 0 0 0 ab3-a3c 0 -ab2c-a2bd -ab2c-a2bd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b4-a2bc ab3-a3c 0 b3c+ab2d 0 0 0 0 0 -b3d+a2cd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b -c -d 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | {3} | a b -c -d 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | | {4} | -ab -b2 -bc -bd -a2b 0 0 ab2 0 0 0 0 abc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b3 ab2 0 0 0 b2c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 bc2 b2c abc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | {4} | -b 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | |{-1, 4} |{0, 4} | | | {3} | 0 0 -a3 -a3 0 0 0 -a2b 0 0 -a2b -a2b -a3 0 0 0 -ab2 -ab2 -a2b 0 0 -a2c -a2c -a2c -a2b -a2d -abc -a2d 0 0 0 0 -abd -abd 0 0 0 -ab2 -ab2 -a2b 0 -b3 -b3 -ab2 0 0 -abc -abc -abc -a2c -ab2 -abd -b2c -abd -a2d -a2c -b2d -b2d -b2c -abc -b3 -ac2+b2d -b2d -abd -abc -acd -acd -bc2 -acd -abd -ad2 -bcd -bcd 0 -ad2 0 0 0 0 -bd2 -bd2 0 0 a3b a3b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a2b2 a2b2 0 0 0 0 0 0 ab3-a3c 0 0 0 0 0 0 0 0 0 0 0 ab3-a3c -a3c 0 0 0 0 0 b4-a2bc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 | {3} | 0 0 a b 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 | | {4} | a2 ab ac ad a3 0 0 -a2b 0 0 0 0 -a2c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -ab2 -a2b 0 0 0 -abc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -ac2 -abc -a2c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | {4} | a 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | | | | | | | | | | | |{-3, 4} |{-2, 4} | | | 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|0 |subquotient (| a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 bc-ad b3-a2c ac2-b2d c3-bd2 a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 |, | bc-ad b3-a2c ac2-b2d c3-bd2 a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 |) |subquotient ({2} | 0 ab2 abd abd 0 a2c a2c ac2 ab2 b3 b2d b2d abd abc abc abc bc2 bc2 ac2 b2c b2c bcd bcd bcd 0 bd2 c3 0 bd2 0 acd acd acd c2d c2d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b2 -ac -bd -c2 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |, {2} | -b2 -ac -bd -c2 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 |)|subquotient ({4} | d ad 0 0 -bd 0 0 0 0 -cd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |, {4} | d a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |) |subquotient ({5} | a b c d |, {5} | a b c d |) | | | | {3} | 0 -ac 0 0 0 -ad -ad 0 -ac -bc 0 0 0 -bd -bd -bd 0 0 0 -c2 -c2 0 0 0 cd 0 0 0 0 0 -d2 -d2 -d2 0 0 0 0 -acd -acd 0 0 0 0 0 0 0 0 0 0 0 0 0 -ac2+b2d 0 -bcd -bcd 0 0 0 0 0 0 -c3+bd2 0 0 0 0 0 0 0 0 c d 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | {3} | c d 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | | {4} | -c -ac 0 0 bc 0 0 0 0 c2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | {4} | -c 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | | | |{-4, 3} |{-3, 3} | {3} | 0 -a2 ac ac 0 -ab -ab ad -a2 -ab bc bc ac -b2 -b2 -b2 bd bd ad -ac -ac c2 c2 c2 bc+ad cd cd 0 cd 0 -bd -bd -bd d2 d2 0 0 -abc-a2d -abc-a2d 0 0 0 0 0 0 0 0 0 0 0 0 0 b3-a2c 0 -b2c-abd -b2c-abd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b -c -d 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | {3} | a b -c -d 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | | {4} | -b -ab 0 0 b2 0 0 0 0 bc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | {4} | -b 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | |{0, 3} | | | | {3} | 0 0 -a2 -a2 0 0 0 -ab 0 0 -ab -ab -a2 0 0 0 -b2 -b2 -ab 0 0 -ac -ac -ac -ab -ad -bc 0 -ad 0 0 0 0 -bd -bd 0 0 a2b a2b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ab2 ab2 0 0 0 0 0 0 b3-a2c 0 0 0 0 0 0 0 0 0 0 a b 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | {3} | 0 0 a b 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | | {4} | a a2 0 0 -ab 0 0 0 0 -ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | {4} | a 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | | | | | | | | | | | |{-2, 3} |{-1, 3} | | 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|0 |0 |subquotient (| a2 ab ac ad b2 bc bd c2 cd d2 bc-ad b3-a2c ac2-b2d c3-bd2 a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 |, | bc-ad b3-a2c ac2-b2d c3-bd2 a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 |) |subquotient ({2} | 0 0 b2 bd 0 bd 0 ac ac c2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -b2 -ac -bd -c2 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |, {2} | -b2 -ac -bd -c2 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 |)|subquotient ({4} | d d a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 |, {4} | d a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 |) | | | | | {3} | 0 0 -c 0 0 0 0 -d -d 0 0 0 -cd -cd 0 0 0 0 0 0 0 0 0 0 0 0 0 c d 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | {3} | c d 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | | {4} | -c -c 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | {4} | -c 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | | |{-4, 2} |{-3, 2} |{-2, 2} | {3} | 0 0 -a c 0 c 0 -b -b d 0 0 -bc-ad -bc-ad 0 0 0 0 0 0 0 0 0 0 0 0 0 a b -c -d 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | {3} | a b -c -d 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | | {4} | -b -b 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | {4} | -b 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | | | | | | {3} | 0 0 0 -a 0 -a 0 0 0 -b 0 0 ab ab 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | {3} | 0 0 a b 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | | {4} | a a 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | {4} | a 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | | | | | | | | | | | |{-1, 2} |{0, 2} | 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|0 |0 |0 |subquotient (| a b c d bc-ad b3-a2c ac2-b2d c3-bd2 a2 ab ac ad b2 bc bd c2 cd d2 |, | bc-ad b3-a2c ac2-b2d c3-bd2 a2 ab ac ad b2 bc bd c2 cd d2 |) |subquotient ({2} | ac b2 c2 bd -b2 -ac -bd -c2 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 |, {2} | -b2 -ac -bd -c2 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 |)| | | | | | {3} | -d -c 0 0 c d 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | {3} | c d 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | | |{-4, 1} |{-3, 1} |{-2, 1} |{-1, 1} | {3} | -b -a d c a b -c -d 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | {3} | a b -c -d 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | | | | | | | {3} | 0 0 -b -a 0 0 a b 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | {3} | 0 0 a b 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | | | | | | | | | | | | |{0, 1} | 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|0 |0 |0 |0 |cokernel | bc-ad b3-a2c ac2-b2d c3-bd2 a b c d | | | | | | | | |{-4, 0} |{-3, 0} |{-2, 0} |{-1, 0} |{0, 0} | 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o8 : Page```