var t;
var x;
var y;
var z;

subject to

cons1:  x^3*y^2+4*x^2*y^2*z-x^2*y*z^2+288*x^2*y^2+207*x^2*y*z+1152*x*y^2*z+
   156*x*y*z^2+x*z^3-3456*x^2*y+20736*x*y^2+19008*x*y*z+82944*y^2*z
   +432*x*z^2- 497664*x*y+62208*x*z+2985984*x = 0;
cons2:  y^3*t^3+4*y^3*t^2-y^2*z*t^2+4*y^2*t^3-48*y^2*t^2-5*y*z*t^2
   +108*y*z*t+ z^2*t+144*z*t-1728*z = 0;
cons3:  -x^2*z^2*t+4*x*z^2*t^2+z^3*t^2+x^3*z+156*x^2*z*t+207*x*z^2*t+
   1152*x*z*t^2+288*z^2*t^2+432*x^2*z+19008*x*z*t-3456*z^2*t+82944*x*t^2+
   20736*z*t^2+62208*x*z-497664*z*t+2985984*z = 0;
cons4:  y^3*t^3-x*y^2*t^2+4*y^3*t^2+4*y^2*t^3-5*x*y^2*t-48*y^2*t^2+x^2*y+
   108*x*y*t+144*x*y-1728*x = 0;

solve;
display t, x, y, z;

# TITLE : cohn2, modular equations for special algebraic number fields

# ROOT COUNTS :

# total degree : 900
# 2-homogeneous Bezout number is 450
#   with partition : {x y z }{t }
# generalized Bezout number : 358
#   based on the set structure :
#      {x }{x z }{x z }{y }{y z }
#      {y z }{y }{y z }{t }{t }{t }
#      {x z }{x t }{z }{x z }{t }
#      {y }{x y }{x y }{t }{t }{t }
# mixed volume : 124

# REFERENCES :

# From 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 explit modular equation in two variables
# for Q[sqrt(3)]", Math. of Comp. 50, pp. 557-568, 1988.

# THE SOLUTIONS :

# 18 4
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -1.23997158246700E+02  -7.54108290873468E+01
#  y :  4.81861750852218E+01   2.85294148295339E+01
#  z : -1.61588268408412E+01   1.78870081595856E+01
#  t : -3.05964654408600E+00   1.82602611365708E-01
# == err :  2.401E-13 = rco :  2.653E-04 = res :  4.265E-06 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  6.02393786358094E+02   2.51345585423244E-88
#  y :  2.08383161251585E+01  -5.10545720390964E-89
#  z :  6.02393786358094E+02  -9.73964143515069E-88
#  t :  2.08383161251585E+01  -6.13636683162216E-89
# == err :  5.985E-13 = rco :  5.275E-06 = res :  5.035E-04 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -5.97960427705686E+01   6.80828237761971E+00
#  y : -4.99290642112624E-01   9.33089973938555E+00
#  z : -5.97960427705686E+01   6.80828237761971E+00
#  t : -4.99290642112627E-01   9.33089973938555E+00
# == err :  1.830E-14 = rco :  5.344E-04 = res :  2.107E-07 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -8.42105246557411E+02  -1.12909149701848E-89
#  y : -7.66022331457492E+00   8.43750439348047E-92
#  z : -1.09615038022715E+01  -3.68182009897330E-91
#  t : -2.18804311393714E+01  -2.87642195232289E-91
# == err :  9.157E-14 = rco :  7.570E-07 = res :  9.507E-06 ==
# solution 5 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -5.97960427705686E+01  -6.80828237761971E+00
#  y : -4.99290642112623E-01  -9.33089973938555E+00
#  z : -5.97960427705686E+01  -6.80828237761971E+00
#  t : -4.99290642112629E-01  -9.33089973938555E+00
# == err :  2.214E-14 = rco :  5.344E-04 = res :  9.424E-08 ==
# solution 6 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  2.30226349318897E+00   2.29757961177032E+02
#  y :  1.17605675035346E+00  -2.87963907718388E+00
#  z :  2.30226349318902E+00  -2.29757961177032E+02
#  t :  1.17605675035346E+00   2.87963907718388E+00
# == err :  4.355E-13 = rco :  5.736E-07 = res :  3.279E-06 ==
# solution 7 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  4.35414647301048E-01  -7.37888274251716E+00
#  y :  1.83176559758234E+00  -4.11273135653901E+00
#  z :  4.35414647301049E-01  -7.37888274251716E+00
#  t :  1.83176559758234E+00  -4.11273135653901E+00
# == err :  1.640E-15 = rco :  2.781E-04 = res :  5.890E-09 ==
# solution 8 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  4.35414647301048E-01   7.37888274251716E+00
#  y :  1.83176559758234E+00   4.11273135653901E+00
#  z :  4.35414647301049E-01   7.37888274251716E+00
#  t :  1.83176559758234E+00   4.11273135653901E+00
# == err :  1.640E-15 = rco :  2.781E-04 = res :  5.890E-09 ==
# solution 9 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  6.02648764055773E+02  -8.10922451376730E+02
#  y :  1.17711982346853E+00   9.71239630969622E-01
#  z :  6.02648764055773E+02  -8.10922451376730E+02
#  t :  1.17711982346853E+00   9.71239630969621E-01
# == err :  4.823E-13 = rco :  1.265E-09 = res :  3.553E-04 ==
# solution 10 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -1.09615038022715E+01  -4.29545678213551E-91
#  y : -2.18804311393714E+01   4.14204761134496E-91
#  z : -8.42105246557411E+02   1.62000084354825E-89
#  t : -7.66022331457492E+00   0.00000000000000E+00
# == err :  1.783E-13 = rco :  7.580E-07 = res :  1.144E-05 ==
# solution 11 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -1.61588268408412E+01   1.78870081595856E+01
#  y : -3.05964654408600E+00   1.82602611365708E-01
#  z : -1.23997158246700E+02  -7.54108290873469E+01
#  t :  4.81861750852218E+01   2.85294148295338E+01
# == err :  2.167E-13 = rco :  3.095E-04 = res :  5.150E-06 ==
# solution 12 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  1.36567442620473E+02   6.73357734978399E+01
#  y : -1.33747712198973E+01  -2.20687866697351E+00
#  z :  1.36567442620473E+02   6.73357734978399E+01
#  t : -1.33747712198973E+01  -2.20687866697351E+00
# == err :  2.057E-13 = rco :  1.822E-05 = res :  2.861E-06 ==
# solution 13 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -1.61588268408412E+01  -1.78870081595856E+01
#  y : -3.05964654408600E+00  -1.82602611365708E-01
#  z : -1.23997158246700E+02   7.54108290873469E+01
#  t :  4.81861750852218E+01  -2.85294148295338E+01
# == err :  2.167E-13 = rco :  3.095E-04 = res :  5.150E-06 ==
# solution 14 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  2.30226349318900E+00  -2.29757961177032E+02
#  y :  1.17605675035346E+00   2.87963907718388E+00
#  z :  2.30226349318901E+00   2.29757961177032E+02
#  t :  1.17605675035346E+00  -2.87963907718388E+00
# == err :  3.868E-13 = rco :  5.736E-07 = res :  5.030E-06 ==
# solution 15 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  1.36567442620473E+02  -6.73357734978399E+01
#  y : -1.33747712198973E+01   2.20687866697351E+00
#  z :  1.36567442620473E+02  -6.73357734978399E+01
#  t : -1.33747712198973E+01   2.20687866697351E+00
# == err :  2.057E-13 = rco :  1.822E-05 = res :  2.861E-06 ==
# solution 16 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -6.71049434640517E+01   7.06909459002873E-89
#  y : -1.21079632432404E+01   2.07409198908829E-89
#  z : -6.71049434640517E+01  -6.28363963558109E-89
#  t : -1.21079632432404E+01  -2.25818299403695E-89
# == err :  5.000E-14 = rco :  3.370E-04 = res :  5.960E-08 ==
# solution 17 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x :  6.02648764055773E+02   8.10922451376730E+02
#  y :  1.17711982346853E+00  -9.71239630969622E-01
#  z :  6.02648764055773E+02   8.10922451376730E+02
#  t :  1.17711982346853E+00  -9.71239630969621E-01
# == err :  5.276E-13 = rco :  1.265E-09 = res :  3.200E-04 ==
# solution 18 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  x : -1.23997158246700E+02   7.54108290873468E+01
#  y :  4.81861750852218E+01  -2.85294148295339E+01
#  z : -1.61588268408412E+01  -1.78870081595856E+01
#  t : -3.05964654408600E+00  -1.82602611365708E-01
# == err :  2.401E-13 = rco :  2.653E-04 = res :  4.265E-06 ==