Pasted

box% sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 9.1, Release Date: 2020-05-20                     │
│ Using Python 3.8.6. Type "help()" for help.                        │
└────────────────────────────────────────────────────────────────────┘
sage: R.<x0,x1,x2,x3,x4> = PolynomialRing(QQ, order='lex')
sage: add = lambda p, q: tuple(pi+qi for pi,qi in zip(p,q))
sage: mul = lambda p, q: tuple(sum(pi*qj for (i,pi) in enumerate(p) for (j,qj) in enumerate(q) if (i+j)%5==k) for k in range(5))
sage: x = (x0,x1,x2,x3,x4)
sage: u1=(x0,x1,x2,x3,x4); u2=(x0,x2,x4,x1,x3); u3=(x0,x3,x1,x4,x2); u4=(x0,x4,x3,x2,x1)
sage: s1=add(u1,add(u2,add(u3,u4)))
sage: s2=add(mul(u1,u2),add(mul(u1,u3),add(mul(u1,u4),add(mul(u2,u3),add(mul(u2,u4),mul(u3,u4))))))
sage: s3=add(mul(u1,mul(u2,u3)),add(mul(u1,mul(u2,u4)),add(mul(u1,mul(u3,u4)),mul(u2,mul(u3,u4)))))
sage: s4=mul(u1,mul(u2,mul(u3,u4)))
sage: ideal([si[0]-si[1] for si in (s1,s2,s3,s4)]).groebner_basis()[-1].factor()
(x3 - x4)^7
sage: