# cohn3 -- modular equations for special algebraic number fields

## Synopsis

• Usage:
cohn3(kk)
• Inputs:
• kk, a ring, the coefficient ring
• Outputs:
• a list, of the polynomials in the system

## Description

This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 72 solutions found in 13.8985 seconds (with a Bezout bound of 1080).

Reference: See the PoSSo test suite. Andre' Galligo and Carlo Traverso. "Practical Determination of the dimension of an algebraic variety", in E. Kaltofen and S.M. Watt, Eds "Computers and Mathematics", pages 46-52, 1989.

H. Cohn. "An explicit modular equation in two variables and Hilbert's Twelfth problem", Math. of Comp. 38, pp. 227-236, 1982.

H. Cohn, J. Deutch. "An explicit modular equation in two variables for Q[sqrt(3)]", Math. of Comp. 50, pp. 557-568, 1988.

 i1 : F = cohn3(QQ) 3 2 2 2 2 2 2 2 2 2 2 o1 = {- x y + 2x y z - x y*z - 144x y - 207x y*z + 288x*y z + 78x*y*z + ------------------------------------------------------------------------ 3 2 2 2 x*z - 3456x y - 5184x*y - 9504x*y*z - 432x*z - 248832x*y + 62208x*z - ------------------------------------------------------------------------ 3 2 2 2 2 3 2 2 3 2 2 2 2985984x, - x z*t + x z t - 6x z*t + 4x z t + 32x t - 72x z*t - ------------------------------------------------------------------------ 2 2 3 2 3 2 2 2 2 2 87x*z t - z t - 8x z - 432x z*t - 414x*z t + 2592x*z*t + 864z t - ------------------------------------------------------------------------ 2 2 2 1728x z - 20736x*z*t + 3456z t - 186624z*t - 124416x*z - 1492992z*t - ------------------------------------------------------------------------ 2 3 2 3 3 3 2 2 2 2 3 2 2985984z, x y*t - 2x*y t + y t + 8x y*t - 12x*y t + 4y t - ------------------------------------------------------------------------ 3 2 3 2 2 2 2 2 3 24x*y*t + 24y t + 20x y*t - 20x*y t - 160x*y*t + 96y t + 128x*t + ------------------------------------------------------------------------ 2 2 3 3 2 3 16x y + 96x*y*t + 2304x*t + 1152x*y + 13824x*t + 27648x, y t - y z*t ------------------------------------------------------------------------ 3 2 2 2 2 3 3 2 2 2 2 2 + 4y t - 2y z*t + 72y t + 71y*z*t + 288y t + 360y*z*t + 6z t + ------------------------------------------------------------------------ 3 3 2 2 2 3 1728y*t - 464z*t + 432y*z*t + 8z t + 6912y*t - 4320z*t + 13824t + ------------------------------------------------------------------------ 2 2 z - 13824z*t + 55296t - 13824z} o1 : List