For those without magma, go to http://magma.maths.usyd.edu.au/calc/

Below is magma code to compute F,G,H. Please report any erros to gwalsh@uottawa.ca

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F

Magma code

R<x,m,a,b>:=PolynomialRing(Rationals(),4);
p:=3*(x-m^2)^2+(2*(a+b)*(x-m^2))+(a*b);
ell:=(x-m^2)^3+(a+b)*(x-m^2)^2+a*b*(x-m^2)+m^6;
F:=p^2-(4*ell)*(-m^2+(a+b)+2*(x-m^2));
F;

x^4 - 6*x^2*m^4 + 4*x^2*m^2*a + 4*x^2*m^2*b - 2*x^2*a*b - 8*x*m^4*a - 8*x*m^4*b + 8*x*m^2*a*b + 9*m^8 - 6*m^4*a*b + a^2*b^2

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G

Magma code

R<x,m,a,b>:=PolynomialRing(Rationals(),4);
S<n>:=PolynomialRing(R);

numerlambda1:=(m^3-m*n);
denomlambda1:=m^2-a;
numerXofPplusR:=numerlambda1^2-((m^2-b)*denomlambda1^2);N1:=numerXofPplusR;
denomXofPplusR:=(m^2-a)^2;D1:=denomXofPplusR;

p:=3*x^2+2*(a+b)*x+(a*b);
ell:=(x)^3+(a+b)*(x)^2+a*b*(x)+m^6;

numerXofdouble:=p^2-(4*ell)*(a+b+2*x);N2:=numerXofdouble;
denomXofdouble:=4*ell;D2:=denomXofdouble;

/*
use N1/D1 = N2/D2 by computing the cross product:
*/

F1:=N1*D2-N2*D1;

/*
replace n^2 by (a+b)m^2-ab:
*/


F2:=(4*x^3*m^2 + 4*x^2*m^2*a + 4*x^2*m^2*b + 4*x*m^2*a*b + 4*m^8)*((a+b)*m^2-a*b)^2 + (-8*x^3*m^4
    - 8*x^2*m^4*a - 8*x^2*m^4*b - 8*x*m^4*a*b - 8*m^10)*n - x^4*m^4 +
    2*x^4*m^2*a - x^4*a^2 + 8*x^3*m^4*a + 4*x^3*m^4*b - 4*x^3*m^2*a^2 -
    8*x^3*m^2*a*b + 4*x^3*a^2*b + 8*x^2*m^4*a^2 + 14*x^2*m^4*a*b + 4*x^2*m^4*b^2
    - 4*x^2*m^2*a^3 - 16*x^2*m^2*a^2*b - 8*x^2*m^2*a*b^2 + 6*x^2*a^3*b +
    4*x^2*a^2*b^2 + 8*x*m^10 - 16*x*m^8*a + 8*x*m^6*a^2 + 8*x*m^4*a^2*b +
    4*x*m^4*a*b^2 - 4*x*m^2*a^3*b - 8*x*m^2*a^2*b^2 + 4*x*a^3*b^2 + 12*m^10*a +
    8*m^10*b - 12*m^8*a^2 - 16*m^8*a*b + 4*m^6*a^3 + 8*m^6*a^2*b - m^4*a^2*b^2 +
    2*m^2*a^3*b^2 - a^4*b^2;

/*
separate the constant term from the multiple of n, square both sides, and replace n^2 by (a+b)m^2-ab
*/

C1:=Coefficient(F2,1);
C0:=Coefficient(F2,0);
G:=-C1^2*((a+b)*m^2-a*b)+C0^2;

G:=R!G;G;

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H

Magma code

R<x,m,a,b>:=PolynomialRing(Rationals(),4);
S<n>:=PolynomialRing(R);

p:=3*x^2+2*(a+b)*x+(a*b);
ell:=(x)^3+(a+b)*(x)^2+a*b*(x)+m^6;

F:=p^2*m^2-4*ell*(m^2+n)^2-(2*x+m^2)*(4*ell)*m^2;


F1:=(-4*x^3 - 4*x^2*a - 4*x^2*b - 4*x*a*b - 4*m^6)*((a+b)*m^2-a*b)^2 + (-8*x^3*m^2 - 8*x^2*m^2*a -
8*x^2*m^2*b - 8*x*m^2*a*b - 8*m^8)*n + x^4*m^2 - 8*x^3*m^4 + 4*x^3*m^2*a +
4*x^3*m^2*b - 8*x^2*m^4*a - 8*x^2*m^4*b + 4*x^2*m^2*a^2 + 6*x^2*m^2*a*b +
4*x^2*m^2*b^2 - 8*x*m^8 - 8*x*m^4*a*b + 4*x*m^2*a^2*b + 4*x*m^2*a*b^2 -
8*m^10 + m^2*a^2*b^2;


C1:=Coefficient(F1,1);C0:=Coefficient(F1,0);

H:=-C1^2*((a+b)*m^2-a*b)+C0^2;H:=R!H;H;