-- M2-Script used to compute the fundamental syzygy between
-- the fundamental invariants of the binary quintic
-- author: Holger Cröni <croeni@epost.de>
-- October 2001


R = ZZ/31991 [a..f] 
-- for an infinte field use:  R = QQ  [a..f]

i4 = -32*c*c*d*d+48*b*d*d*d+48*c*c*c*e-76*b*c*d*e-12*a*d*d*e+9*b*b*e*e+16*a*c*e*e-12*b*c*c*f+16*b*b*d*f+4*a*c*d*f-10*a*b*e*f+1*a*a*f*f

i8 = 8*c*c*c*c*d*d*d*d-24*b*c*c*d*d*d*d*d+18*b*b*d*d*d*d*d*d+0*a*c*d*d*d*d*d*d-24*c*c*c*c*c*d*d*e+74*b*c*c*c*d*d*d*e-57*b*b*c*d*d*d*d*e+6*a*c*c*d*d*d*d*e-9*a*b*d*d*d*d*d*e+18*c*c*c*c*c*c*e*e-57*b*c*c*c*c*d*e*e+25*b*b*c*c*d*d*e*e+8*a*c*c*c*d*d*e*e+38*b*b*b*d*d*d*e*e-30*a*b*c*d*d*d*e*e+18*a*a*d*d*d*d*e*e+38*b*b*c*c*c*e*e*e-38*a*c*c*c*c*e*e*e-81*b*b*b*c*d*e*e*e+106*a*b*c*c*d*e*e*e+3*a*b*b*d*d*e*e*e-48*a*a*c*d*d*e*e*e+27*b*b*b*b*e*e*e*e-54*a*b*b*c*e*e*e*e+22*a*a*c*c*e*e*e*e+15*a*a*b*d*e*e*e*e-2*a*a*a*e*e*e*e*e+0*c*c*c*c*c*c*d*f+6*b*c*c*c*c*d*d*f+8*b*b*c*c*d*d*d*f-42*a*c*c*c*d*d*d*f-38*b*b*b*d*d*d*d*f+93*a*b*c*d*d*d*d*f-27*a*a*d*d*d*d*d*f-9*b*c*c*c*c*c*e*f-30*b*b*c*c*c*d*e*f+93*a*c*c*c*c*d*e*f+106*b*b*b*c*d*d*e*f-220*a*b*c*c*d*d*e*f-18*a*b*b*d*d*d*e*f+78*a*a*c*d*d*d*e*f+3*b*b*b*c*c*e*e*f-18*a*b*c*c*c*e*e*f-54*b*b*b*b*d*e*e*f+133*a*b*b*c*d*e*e*f-34*a*a*c*c*d*e*e*f-30*a*a*b*d*d*e*e*f+0*a*b*b*b*e*e*e*f-5*a*a*b*c*e*e*e*f+5*a*a*a*d*e*e*e*f+18*b*b*c*c*c*c*f*f-27*a*c*c*c*c*c*f*f-48*b*b*b*c*c*d*f*f+78*a*b*c*c*c*d*f*f+22*b*b*b*b*d*d*f*f-34*a*b*b*c*d*d*f*f-21*a*a*c*c*d*d*f*f+12*a*a*b*d*d*d*f*f+15*b*b*b*b*c*e*f*f-30*a*b*b*c*c*e*f*f+12*a*a*c*c*c*e*f*f-5*a*b*b*b*d*e*f*f+11*a*a*b*c*d*e*f*f-3*a*a*a*d*d*e*f*f+1*a*a*b*b*e*e*f*f-1*a*a*a*c*e*e*f*f-2*b*b*b*b*b*f*f*f+5*a*b*b*b*c*f*f*f-3*a*a*b*c*c*f*f*f-1*a*a*b*b*d*f*f*f+1*a*a*a*c*d*f*f*f+0*a*a*a*b*e*f*f*f+0*a*a*a*a*f*f*f*f


i12 = -16*c*c*c*c*c*c*d*d*d*d*d*d+72*b*c*c*c*c*d*d*d*d*d*d*d-83*b*b*c*c*d*d*d*d*d*d*d*d-40*a*c*c*c*d*d*d*d*d*d*d*d+4*b*b*b*d*d*d*d*d*d*d*d*d+90*a*b*c*d*d*d*d*d*d*d*d*d-27*a*a*d*d*d*d*d*d*d*d*d*d+72*c*c*c*c*c*c*c*d*d*d*d*e-330*b*c*c*c*c*c*d*d*d*d*d*e+354*b*b*c*c*c*d*d*d*d*d*d*e+222*a*c*c*c*c*d*d*d*d*d*d*e+56*b*b*b*c*d*d*d*d*d*d*d*e-506*a*b*c*c*d*d*d*d*d*d*d*e-48*a*b*b*d*d*d*d*d*d*d*d*e+180*a*a*c*d*d*d*d*d*d*d*d*e-83*c*c*c*c*c*c*c*c*d*d*e*e+354*b*c*c*c*c*c*c*d*d*d*e*e-139*b*b*c*c*c*c*d*d*d*d*e*e-392*a*c*c*c*c*c*d*d*d*d*e*e-550*b*b*b*c*c*d*d*d*d*d*e*e+804*a*b*c*c*c*d*d*d*d*d*e*e-4*b*b*b*b*d*d*d*d*d*d*e*e+516*a*b*b*c*d*d*d*d*d*d*e*e-428*a*a*c*c*d*d*d*d*d*d*e*e-78*a*a*b*d*d*d*d*d*d*d*e*e+4*c*c*c*c*c*c*c*c*c*e*e*e+56*b*c*c*c*c*c*c*c*d*e*e*e-550*b*b*c*c*c*c*c*d*d*e*e*e+220*a*c*c*c*c*c*c*d*d*e*e*e+866*b*b*b*c*c*c*d*d*d*e*e*e-206*a*b*c*c*c*c*d*d*d*e*e*e+246*b*b*b*b*c*d*d*d*d*e*e*e-1246*a*b*b*c*c*d*d*d*d*e*e*e+440*a*a*c*c*c*d*d*d*d*e*e*e-138*a*b*b*b*d*d*d*d*d*e*e*e+294*a*a*b*c*d*d*d*d*d*e*e*e+14*a*a*a*d*d*d*d*d*d*e*e*e-4*b*b*c*c*c*c*c*c*e*e*e*e-16*a*c*c*c*c*c*c*c*e*e*e*e+246*b*b*b*c*c*c*c*d*e*e*e*e-136*a*b*c*c*c*c*c*d*e*e*e*e-713*b*b*b*b*c*c*d*d*e*e*e*e+652*a*b*b*c*c*c*d*d*e*e*e*e-194*a*a*c*c*c*c*d*d*e*e*e*e-32*b*b*b*b*b*d*d*d*e*e*e*e+590*a*b*b*b*c*d*d*d*e*e*e*e-298*a*a*b*c*c*d*d*d*e*e*e*e-39*a*a*b*b*d*d*d*d*e*e*e*e-56*a*a*a*c*d*d*d*d*e*e*e*e-32*b*b*b*b*c*c*c*e*e*e*e*e+4*a*b*b*c*c*c*c*e*e*e*e*e+24*a*a*c*c*c*c*c*e*e*e*e*e+234*b*b*b*b*b*c*d*e*e*e*e*e-308*a*b*b*b*c*c*d*e*e*e*e*e+104*a*a*b*c*c*c*d*e*e*e*e*e-102*a*b*b*b*b*d*d*e*e*e*e*e-6*a*a*b*b*c*d*d*e*e*e*e*e+60*a*a*a*c*c*d*d*e*e*e*e*e+22*a*a*a*b*d*d*d*e*e*e*e*e-27*b*b*b*b*b*b*e*e*e*e*e*e+36*a*b*b*b*b*c*e*e*e*e*e*e+4*a*a*b*b*c*c*e*e*e*e*e*e-16*a*a*a*c*c*c*e*e*e*e*e*e+30*a*a*b*b*b*d*e*e*e*e*e*e-24*a*a*a*b*c*d*e*e*e*e*e*e-3*a*a*a*a*d*d*e*e*e*e*e*e-4*a*a*a*b*b*e*e*e*e*e*e*e+4*a*a*a*a*c*e*e*e*e*e*e*e-40*c*c*c*c*c*c*c*c*d*d*d*f+222*b*c*c*c*c*c*c*d*d*d*d*f-392*b*b*c*c*c*c*d*d*d*d*d*f-106*a*c*c*c*c*c*d*d*d*d*d*f+220*b*b*b*c*c*d*d*d*d*d*d*f+342*a*b*c*c*c*d*d*d*d*d*d*f-16*b*b*b*b*d*d*d*d*d*d*d*f-224*a*b*b*c*d*d*d*d*d*d*d*f-78*a*a*c*c*d*d*d*d*d*d*d*f+72*a*a*b*d*d*d*d*d*d*d*d*f+90*c*c*c*c*c*c*c*c*c*d*e*f-506*b*c*c*c*c*c*c*c*d*d*e*f+804*b*b*c*c*c*c*c*d*d*d*e*f+342*a*c*c*c*c*c*c*d*d*d*e*f-206*b*b*b*c*c*c*d*d*d*d*e*f-1078*a*b*c*c*c*c*d*d*d*d*e*f-136*b*b*b*b*c*d*d*d*d*d*e*f+498*a*b*b*c*c*d*d*d*d*d*e*f+324*a*a*c*c*c*d*d*d*d*d*e*f+128*a*b*b*b*d*d*d*d*d*d*e*f-242*a*a*b*c*d*d*d*d*d*d*e*f-18*a*a*a*d*d*d*d*d*d*d*e*f-48*b*c*c*c*c*c*c*c*c*e*e*f+516*b*b*c*c*c*c*c*c*d*e*e*f-224*a*c*c*c*c*c*c*c*d*e*e*f-1246*b*b*b*c*c*c*c*d*d*e*e*f+498*a*b*c*c*c*c*c*d*d*e*e*f+652*b*b*b*b*c*c*d*d*d*e*e*f+714*a*b*b*c*c*c*d*d*d*e*e*f-394*a*a*c*c*c*c*d*d*d*e*e*f+4*b*b*b*b*b*d*d*d*d*e*e*f-674*a*b*b*b*c*d*d*d*d*e*e*f+42*a*a*b*c*c*d*d*d*d*e*e*f+90*a*a*b*b*d*d*d*d*d*e*e*f+70*a*a*a*c*d*d*d*d*d*e*e*f-138*b*b*b*c*c*c*c*c*e*e*e*f+128*a*b*c*c*c*c*c*c*e*e*e*f+590*b*b*b*b*c*c*c*d*e*e*e*f-674*a*b*b*c*c*c*c*d*e*e*e*f+164*a*a*c*c*c*c*c*d*e*e*e*f-308*b*b*b*b*b*c*d*d*e*e*e*f-42*a*b*b*b*c*c*d*d*e*e*e*f+170*a*a*b*c*c*c*d*d*e*e*e*f+204*a*b*b*b*b*d*d*d*e*e*e*f+2*a*a*b*b*c*d*d*d*e*e*e*f-46*a*a*a*c*c*d*d*d*e*e*e*f-50*a*a*a*b*d*d*d*d*e*e*e*f-102*b*b*b*b*b*c*c*e*e*e*e*f+204*a*b*b*b*c*c*c*e*e*e*e*f-112*a*a*b*c*c*c*c*e*e*e*e*f+36*b*b*b*b*b*b*d*e*e*e*e*f-38*a*b*b*b*b*c*d*e*e*e*e*f+48*a*a*b*b*c*c*d*e*e*e*e*f-16*a*a*a*c*c*c*d*e*e*e*e*f-60*a*a*b*b*b*d*d*e*e*e*e*f+30*a*a*a*b*c*d*d*e*e*e*e*f+10*a*a*a*a*d*d*d*e*e*e*e*f+18*a*b*b*b*b*b*e*e*e*e*e*f-50*a*a*b*b*b*c*e*e*e*e*e*f+32*a*a*a*b*c*c*e*e*e*e*e*f+14*a*a*a*b*b*d*e*e*e*e*e*f-14*a*a*a*a*c*d*e*e*e*e*e*f+0*a*a*a*a*b*e*e*e*e*e*e*f-27*c*c*c*c*c*c*c*c*c*c*f*f+180*b*c*c*c*c*c*c*c*c*d*f*f-428*b*b*c*c*c*c*c*c*d*d*f*f-78*a*c*c*c*c*c*c*c*d*d*f*f+440*b*b*b*c*c*c*c*d*d*d*f*f+324*a*b*c*c*c*c*c*d*d*d*f*f-194*b*b*b*b*c*c*d*d*d*d*f*f-394*a*b*b*c*c*c*d*d*d*d*f*f-63*a*a*c*c*c*c*d*d*d*d*f*f+24*b*b*b*b*b*d*d*d*d*d*f*f+164*a*b*b*b*c*d*d*d*d*d*f*f+108*a*a*b*c*c*d*d*d*d*d*f*f-62*a*a*b*b*d*d*d*d*d*d*f*f+6*a*a*a*c*d*d*d*d*d*d*f*f-78*b*b*c*c*c*c*c*c*c*e*f*f+72*a*c*c*c*c*c*c*c*c*e*f*f+294*b*b*b*c*c*c*c*c*d*e*f*f-242*a*b*c*c*c*c*c*c*d*e*f*f-298*b*b*b*b*c*c*c*d*d*e*f*f+42*a*b*b*c*c*c*c*d*d*e*f*f+108*a*a*c*c*c*c*c*d*d*e*f*f+104*b*b*b*b*b*c*d*d*d*e*f*f+170*a*b*b*b*c*c*d*d*d*e*f*f-82*a*a*b*c*c*c*d*d*d*e*f*f-112*a*b*b*b*b*d*d*d*d*e*f*f+48*a*a*b*b*c*d*d*d*d*e*f*f-54*a*a*a*c*c*d*d*d*d*e*f*f+28*a*a*a*b*d*d*d*d*d*e*f*f-39*b*b*b*b*c*c*c*c*e*e*f*f+90*a*b*b*c*c*c*c*c*e*e*f*f-62*a*a*c*c*c*c*c*c*e*e*f*f-6*b*b*b*b*b*c*c*d*e*e*f*f+2*a*b*b*b*c*c*c*d*e*e*f*f+48*a*a*b*c*c*c*c*d*e*e*f*f+4*b*b*b*b*b*b*d*d*e*e*f*f+48*a*b*b*b*b*c*d*d*e*e*f*f-168*a*a*b*b*c*c*d*d*e*e*f*f+50*a*a*a*c*c*c*d*d*e*e*f*f+14*a*a*b*b*b*d*d*d*e*e*f*f+30*a*a*a*b*c*d*d*d*e*e*f*f-11*a*a*a*a*d*d*d*d*e*e*f*f+30*b*b*b*b*b*b*c*e*e*e*f*f-60*a*b*b*b*b*c*c*e*e*e*f*f+14*a*a*b*b*c*c*c*e*e*e*f*f+16*a*a*a*c*c*c*c*e*e*e*f*f-50*a*b*b*b*b*b*d*e*e*e*f*f+132*a*a*b*b*b*c*d*e*e*e*f*f-82*a*a*a*b*c*c*d*e*e*e*f*f-16*a*a*a*b*b*d*d*e*e*e*f*f+16*a*a*a*a*c*d*d*e*e*e*f*f+1*a*a*b*b*b*b*e*e*e*e*f*f-2*a*a*a*b*b*c*e*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i = ideal (i4,i8,i12,i18)
S = ZZ/31991[x,y,z,u,Degrees => {{4},{8},{12},{18}}]
phi = map (R, S, {i4,i8,i12,i18})
timing j = kernel (phi, SubringLimit => 1)
-- for a complete computation use:  timing j = kernel phi