/*
30/10/2006
Here's the Humbert component of discriminant 12 (degree 16) I found:
It's the highest degree relation so far computed (next being discriminant 9
having degree 24).
It was found by first computing a solution mod 3 and "observing" that the
maximum exponent appearing in the monomials is 8, cutting down the size of
the linear algebra problem far enough for the rational solution to be found
in a "finite amount of time" (1066.453 seconds to be precise).
*/

//R<e1,e2,e3>:=PolynomialRing(Rationals(),3);
h12:=

e1^8*e2^4*e3^4 - 4*e1^8*e2^4*e3^3 + 6*e1^8*e2^4*e3^2 - 4*e1^8*e2^4*e3 +
    e1^8*e2^4 - 4*e1^7*e2^5*e3^4 - 16*e1^7*e2^5*e3^3 + 40*e1^7*e2^5*e3^2 -
    16*e1^7*e2^5*e3 - 4*e1^7*e2^5 + 160*e1^7*e2^4*e3^4 - 160*e1^7*e2^4*e3^3 -
    160*e1^7*e2^4*e3^2 + 160*e1^7*e2^4*e3 - 132*e1^7*e2^3*e3^5 -
    272*e1^7*e2^3*e3^4 + 808*e1^7*e2^3*e3^3 - 272*e1^7*e2^3*e3^2 -
    132*e1^7*e2^3*e3 + 384*e1^7*e2^2*e3^5 - 384*e1^7*e2^2*e3^4 -
    384*e1^7*e2^2*e3^3 + 384*e1^7*e2^2*e3^2 - 256*e1^7*e2*e3^5 +
    512*e1^7*e2*e3^4 - 256*e1^7*e2*e3^3 + 6*e1^6*e2^6*e3^4 + 40*e1^6*e2^6*e3^3 +

    164*e1^6*e2^6*e3^2 + 40*e1^6*e2^6*e3 + 6*e1^6*e2^6 - 160*e1^6*e2^5*e3^4 -
    352*e1^6*e2^5*e3^3 - 352*e1^6*e2^5*e3^2 - 160*e1^6*e2^5*e3 -
    272*e1^6*e2^4*e3^5 + 1344*e1^6*e2^4*e3^4 - 608*e1^6*e2^4*e3^3 +
    1344*e1^6*e2^4*e3^2 - 272*e1^6*e2^4*e3 + 384*e1^6*e2^3*e3^6 -
    416*e1^6*e2^3*e3^5 - 480*e1^6*e2^3*e3^4 - 480*e1^6*e2^3*e3^3 -
    416*e1^6*e2^3*e3^2 + 384*e1^6*e2^3*e3 - 762*e1^6*e2^2*e3^6 +
    1064*e1^6*e2^2*e3^5 - 348*e1^6*e2^2*e3^4 + 1064*e1^6*e2^2*e3^3 -
    762*e1^6*e2^2*e3^2 + 384*e1^6*e2*e3^6 - 384*e1^6*e2*e3^5 - 384*e1^6*e2*e3^4

    + 384*e1^6*e2*e3^3 - 4*e1^5*e2^7*e3^4 - 16*e1^5*e2^7*e3^3 +
    40*e1^5*e2^7*e3^2 - 16*e1^5*e2^7*e3 - 4*e1^5*e2^7 - 160*e1^5*e2^6*e3^4 -
    352*e1^5*e2^6*e3^3 - 352*e1^5*e2^6*e3^2 - 160*e1^5*e2^6*e3 +
    808*e1^5*e2^5*e3^5 - 608*e1^5*e2^5*e3^4 + 3696*e1^5*e2^5*e3^3 -
    608*e1^5*e2^5*e3^2 + 808*e1^5*e2^5*e3 - 384*e1^5*e2^4*e3^6 -
    480*e1^5*e2^4*e3^5 - 2208*e1^5*e2^4*e3^4 - 2208*e1^5*e2^4*e3^3 -
    480*e1^5*e2^4*e3^2 - 384*e1^5*e2^4*e3 - 256*e1^5*e2^3*e3^7 +
    1064*e1^5*e2^3*e3^6 - 608*e1^5*e2^3*e3^5 + 3696*e1^5*e2^3*e3^4 -
    608*e1^5*e2^3*e3^3 + 1064*e1^5*e2^3*e3^2 - 256*e1^5*e2^3*e3 +
    384*e1^5*e2^2*e3^7 - 416*e1^5*e2^2*e3^6 - 480*e1^5*e2^2*e3^5 -
    480*e1^5*e2^2*e3^4 - 416*e1^5*e2^2*e3^3 + 384*e1^5*e2^2*e3^2 -
    132*e1^5*e2*e3^7 - 272*e1^5*e2*e3^6 + 808*e1^5*e2*e3^5 - 272*e1^5*e2*e3^4 -

    132*e1^5*e2*e3^3 + e1^4*e2^8*e3^4 - 4*e1^4*e2^8*e3^3 + 6*e1^4*e2^8*e3^2 -
    4*e1^4*e2^8*e3 + e1^4*e2^8 + 160*e1^4*e2^7*e3^4 - 160*e1^4*e2^7*e3^3 -
    160*e1^4*e2^7*e3^2 + 160*e1^4*e2^7*e3 - 272*e1^4*e2^6*e3^5 +
    1344*e1^4*e2^6*e3^4 - 608*e1^4*e2^6*e3^3 + 1344*e1^4*e2^6*e3^2 -
    272*e1^4*e2^6*e3 - 384*e1^4*e2^5*e3^6 - 480*e1^4*e2^5*e3^5 -
    2208*e1^4*e2^5*e3^4 - 2208*e1^4*e2^5*e3^3 - 480*e1^4*e2^5*e3^2 -
    384*e1^4*e2^5*e3 + 512*e1^4*e2^4*e3^7 - 348*e1^4*e2^4*e3^6 +
    3696*e1^4*e2^4*e3^5 + 1496*e1^4*e2^4*e3^4 + 3696*e1^4*e2^4*e3^3 -
    348*e1^4*e2^4*e3^2 + 512*e1^4*e2^4*e3 - 384*e1^4*e2^3*e3^7 -
    480*e1^4*e2^3*e3^6 - 2208*e1^4*e2^3*e3^5 - 2208*e1^4*e2^3*e3^4 -
    480*e1^4*e2^3*e3^3 - 384*e1^4*e2^3*e3^2 - 272*e1^4*e2^2*e3^7 +
    1344*e1^4*e2^2*e3^6 - 608*e1^4*e2^2*e3^5 + 1344*e1^4*e2^2*e3^4 -
    272*e1^4*e2^2*e3^3 + 160*e1^4*e2*e3^7 - 160*e1^4*e2*e3^6 - 160*e1^4*e2*e3^5

    + 160*e1^4*e2*e3^4 + e1^4*e3^8 - 4*e1^4*e3^7 + 6*e1^4*e3^6 - 4*e1^4*e3^5 +
    e1^4*e3^4 - 132*e1^3*e2^7*e3^5 - 272*e1^3*e2^7*e3^4 + 808*e1^3*e2^7*e3^3 -
    272*e1^3*e2^7*e3^2 - 132*e1^3*e2^7*e3 + 384*e1^3*e2^6*e3^6 -
    416*e1^3*e2^6*e3^5 - 480*e1^3*e2^6*e3^4 - 480*e1^3*e2^6*e3^3 -
    416*e1^3*e2^6*e3^2 + 384*e1^3*e2^6*e3 - 256*e1^3*e2^5*e3^7 +
    1064*e1^3*e2^5*e3^6 - 608*e1^3*e2^5*e3^5 + 3696*e1^3*e2^5*e3^4 -
    608*e1^3*e2^5*e3^3 + 1064*e1^3*e2^5*e3^2 - 256*e1^3*e2^5*e3 -
    384*e1^3*e2^4*e3^7 - 480*e1^3*e2^4*e3^6 - 2208*e1^3*e2^4*e3^5 -
    2208*e1^3*e2^4*e3^4 - 480*e1^3*e2^4*e3^3 - 384*e1^3*e2^4*e3^2 +
    808*e1^3*e2^3*e3^7 - 608*e1^3*e2^3*e3^6 + 3696*e1^3*e2^3*e3^5 -
    608*e1^3*e2^3*e3^4 + 808*e1^3*e2^3*e3^3 - 160*e1^3*e2^2*e3^7 -
    352*e1^3*e2^2*e3^6 - 352*e1^3*e2^2*e3^5 - 160*e1^3*e2^2*e3^4 -
    4*e1^3*e2*e3^8 - 16*e1^3*e2*e3^7 + 40*e1^3*e2*e3^6 - 16*e1^3*e2*e3^5 -
    4*e1^3*e2*e3^4 + 384*e1^2*e2^7*e3^5 - 384*e1^2*e2^7*e3^4 -
    384*e1^2*e2^7*e3^3 + 384*e1^2*e2^7*e3^2 - 762*e1^2*e2^6*e3^6 +
    1064*e1^2*e2^6*e3^5 - 348*e1^2*e2^6*e3^4 + 1064*e1^2*e2^6*e3^3 -
    762*e1^2*e2^6*e3^2 + 384*e1^2*e2^5*e3^7 - 416*e1^2*e2^5*e3^6 -
    480*e1^2*e2^5*e3^5 - 480*e1^2*e2^5*e3^4 - 416*e1^2*e2^5*e3^3 +
    384*e1^2*e2^5*e3^2 - 272*e1^2*e2^4*e3^7 + 1344*e1^2*e2^4*e3^6 -
    608*e1^2*e2^4*e3^5 + 1344*e1^2*e2^4*e3^4 - 272*e1^2*e2^4*e3^3 -
    160*e1^2*e2^3*e3^7 - 352*e1^2*e2^3*e3^6 - 352*e1^2*e2^3*e3^5 -
    160*e1^2*e2^3*e3^4 + 6*e1^2*e2^2*e3^8 + 40*e1^2*e2^2*e3^7 +
    164*e1^2*e2^2*e3^6 + 40*e1^2*e2^2*e3^5 + 6*e1^2*e2^2*e3^4 - 256*e1*e2^7*e3^5

    + 512*e1*e2^7*e3^4 - 256*e1*e2^7*e3^3 + 384*e1*e2^6*e3^6 - 384*e1*e2^6*e3^5

    - 384*e1*e2^6*e3^4 + 384*e1*e2^6*e3^3 - 132*e1*e2^5*e3^7 - 272*e1*e2^5*e3^6

    + 808*e1*e2^5*e3^5 - 272*e1*e2^5*e3^4 - 132*e1*e2^5*e3^3 + 160*e1*e2^4*e3^7

    - 160*e1*e2^4*e3^6 - 160*e1*e2^4*e3^5 + 160*e1*e2^4*e3^4 - 4*e1*e2^3*e3^8 -

    16*e1*e2^3*e3^7 + 40*e1*e2^3*e3^6 - 16*e1*e2^3*e3^5 - 4*e1*e2^3*e3^4 +
    e2^4*e3^8 - 4*e2^4*e3^7 + 6*e2^4*e3^6 - 4*e2^4*e3^5 + e2^4*e3^4
;