//The formulas below are written in a form that, with minor modifications, should be
//readable by most computer algebra software.
//In for instance magma, one might want to predefine
//  base:=Rationals();
//  L<r,s,t>:=RationalFunctionField(base,3);
//  _<x>:=PolynomialRing(L);
//  _<k1,k2,k3,k4>:=PolynomialRing(L,4);
//
// or predefine the values r,s,t to special values of interest prior to
// evaluating the assignments defined in this file.


//Expressions Gi, lambdai, Hi (for i=1,2,3,4) such that
//Frst=Gi^2+lambda_i*H_i^3
//is a sextic polynomial in x, independent of the choice of i.
//The genus 2 curve
//  Crst: y^2 = Frst(x)
//has an isotropic (3,3) level structure marked on it.
//See Theorem 6

H1:=x^2 + r*x + t;
lambda1:=4*s;
G1:=(-s*t + s - 1)*x^3 + (-3*s*t + 3*s*r)*x^2 + (-3*s*t*r + 3*s*r^2)*x - s*t^2 + s*r^3 + t;
H2:=x^2 + x + r;
lambda2:=4*s*t;
G2:=(-s*t + s + 1)*x^3 + (-3*s*t + 3*s*r)*x^2 + (-3*s*t*r + 3*s*r^2)*x - s*t^2 + s*r^3 - t;
H3:=s*x^2 + (-s*t + 2*s*r - 1)*x + s*r^2;
lambda3:=4*t/(s^2*t^2 + 2*s*t + 1);
G3:=(s^2*t^2 - s^2*t + 2*s*t + s + 1)/(s*t + 1)*x^3 + (3*s^2*t^2 - 3*s^2*t*r + 3*s*t + 3*s*r)/(s*t + 1)*x^2 + (3*s^2*t^2*r - 3*s^2*t*r^2 + 3*s*t*r + 3*s*r^2)/(s*t + 1)*x + (s^2*t^3 - s^2*t*r^3 + 2*s*t^2 + s*r^3 + t)/(s*t + 1);
H4:=(s*t*r - s*t - s*r^2 + s*r + r)*x^2 + (s*t^2 - s*t*r - s*t - s*r^3 + 2*s*r^2 + t)*x + s*t^2 - s*t*r^2 - s*t*r + s*r^3 + t;
lambda4:=4*s*t/(s^2*t^4 - 6*s^2*t^3*r + 2*s^2*t^3 + 2*s^2*t^2*r^3 + 9*s^2*t^2*r^2 - 6*s^2*t^2*r + s^2*t^2 - 6*s^2*t*r^4 + 2*s^2*t*r^3 + s^2*r^6 + 2*s*t^3 - 6*s*t^2*r + 2*s*t^2 + 2*s*t*r^3 + t^2);
G4:=(s^2*t^3 - 3*s^2*t^2*r - s^2*t*r^3 + 6*s^2*t*r^2 - 3*s^2*t*r + s^2*t - s^2*r^3 + 2*s*t^2 - 3*s*t*r + 2*s*t - s*r^3 + t)/(s*t^2 - 3*s*t*r + s*t + s*r^3 + t)*x^3 + (3*s^2*t^3 - 6*s^2*t^2*r^2 - 3*s^2*t^2 + 9*s^2*t*r^3 - 3*s^2*t*r^2 + 3*s^2*t*r - 3*s^2*r^4 + 3*s*t^2 - 6*s*t*r^2 + 3*s*t*r)/(s*t^2 - 3*s*t*r + s*t + s*r^3 + t)*x^2 + (-3*s^2*t^3*r + 6*s^2*t^3 - 9*s^2*t^2*r + 3*s^2*t*r^4 + 3*s^2*t*r^3 + 3*s^2*t*r^2 - 3*s^2*r^5 - 3*s*t^2*r + 6*s*t^2 - 3*s*t*r^2)/(s*t^2 - 3*s*t*r + s*t + s*r^3 + t)*x + (-s^2*t^4 + 3*s^2*t^3*r + s^2*t^3 - 6*s^2*t^2*r^2 + 3*s^2*t*r^4 + s^2*t*r^3 - s^2*r^6 - 2*s*t^3 + 3*s*t^2*r + s*t^2 - 2*s*t*r^3 - t^2)/(s*t^2 - 3*s*t*r + s*t + s*r^3 + t);

// factors deltaj (j=1,...,7) occurring in Disc(Frst)
delta1:=s;
delta2:=t;
delta3:=s*t + 1;
delta4:=r^3 - 3*r*t + t^2 + t;
delta5:=r^3*s - 3*r*s*t + s*t^2 + s*t + t;
delta6:=r^3*s^2 - 3*r*s^2*t - 3*r*s + s^2*t^2 + s^2*t + 2*s*t + s + 1;
delta7:=r^3*s^2*t + r^3*s - 3*r*s^2*t^2 - 3*r*s*t + s^2*t^3 + s^2*t^2 + 2*s*t^2 + t;

Delta:=r^6*s^2 - 6*r^4*s^2*t - 3*r^4*s + 2*r^3*s^2*t^2 + 2*r^3*s^2*t + 3*r^3*s*t + r^3*s + r^3 + 9*r^2*s^2*t^2 + 6*r^2*s*t - 6*r*s^2*t^3 - 6*r*s^2*t^2 - 9*r*s*t^2 - 3*r*s*t - 3*r*t + s^2*t^4 + 2*s^2*t^3 + s^2*t^2 + 2*s*t^3 + 3*s*t^2 + t^2 + t;

//Expressions Gti, lti, Hti (for i=1,2,3,4) such that
//Ft_rst=Gti^2+lti*Hti^3
//is a sextic polynomial in x independent of the choice of i
//The genus 2 curve
//  Ct_rst: y^2=-3*Ft_rst(x)
//has an isotropic mu3*mu3 level structure marked on it.
//Furthermore, thwe jacobians of Crst and Ct_rst are isogenous and the
//numbering i=1,2,3,4 reflects the relation between their level structures.
//See Theorems 9, 13, and Lemma 15.

Ht1:=(s^2*t^3 - 4*s^2*t^2*r + s^2*t^2 + 6*s^2*t*r^2 - 4*s^2*t*r + s^2*t - s^2*r^4 + 2*s*t^2 - 5*s*t*r + s*t + 3*s*r^2 - s*r + t - r)*x^2 + (s^2*t^3 - 3*s^2*t^2*r^2 + s^2*t^2*r - s^2*t^2 + 6*s^2*t*r^3 - 5*s^2*t*r^2 + 2*s^2*t*r - s^2*r^5 + 2*s*t^2 - 4*s*t*r^2 + s*t*r - s*t + 3*s*r^3 - s*r^2 + t - r^2)*x - s^2*t^3*r + s^2*t^3 + 2*s^2*t^2*r^2 - 2*s^2*t^2*r - s^2*t*r^3 + s^2*t*r^2 - 2*s*t^2*r + 2*s*t^2 + 2*s*t*r^2 - 2*s*t*r - t*r + t;
lt1:=(4*s^3*t^4 - 24*s^3*t^3*r + 8*s^3*t^3 + 8*s^3*t^2*r^3 + 36*s^3*t^2*r^2 - 24*s^3*t^2*r + 4*s^3*t^2 - 24*s^3*t*r^4 + 8*s^3*t*r^3 + 4*s^3*r^6 + 8*s^2*t^3 - 36*s^2*t^2*r + 12*s^2*t^2 + 12*s^2*t*r^3 + 24*s^2*t*r^2 - 12*s^2*t*r - 12*s^2*r^4 + 4*s^2*r^3 + 4*s*t^2 - 12*s*t*r + 4*s*t + 4*s*r^3)/(s^4*t^4 - 6*s^4*t^3*r + 2*s^4*t^3 + 2*s^4*t^2*r^3 + 9*s^4*t^2*r^2 - 6*s^4*t^2*r + s^4*t^2 - 6*s^4*t*r^4 + 2*s^4*t*r^3 + s^4*r^6 + 4*s^3*t^3 - 18*s^3*t^2*r + 6*s^3*t^2 + 4*s^3*t*r^3 + 18*s^3*t*r^2 - 12*s^3*t*r + 2*s^3*t - 6*s^3*r^4 + 2*s^3*r^3 + 6*s^2*t^2 - 18*s^2*t*r + 6*s^2*t + 2*s^2*r^3 + 9*s^2*r^2 - 6*s^2*r + s^2 + 4*s*t - 6*s*r + 2*s + 1);
Gt1:=(-s^5*t^7 + 9*s^5*t^6*r - 2*s^5*t^6 - s^5*t^5*r^3 - 33*s^5*t^5*r^2 + 15*s^5*t^5*r - 2*s^5*t^5 + 6*s^5*t^4*r^4 + 64*s^5*t^4*r^3 - 48*s^5*t^4*r^2 + 15*s^5*t^4*r - 2*s^5*t^4 + s^5*t^3*r^6 - 21*s^5*t^3*r^5 - 54*s^5*t^3*r^4 + 64*s^5*t^3*r^3 - 33*s^5*t^3*r^2 + 9*s^5*t^3*r - s^5*t^3 - 3*s^5*t^2*r^7 + 40*s^5*t^2*r^6 - 21*s^5*t^2*r^5 + 6*s^5*t^2*r^4 - s^5*t^2*r^3 + s^5*t*r^9 - 6*s^5*t*r^8 - 3*s^5*t*r^7 + s^5*t*r^6 + s^5*r^9 - 3*s^4*t^6 + 21*s^4*t^5*r - 3*s^4*t^5 - 2*s^4*t^4*r^3 - 54*s^4*t^4*r^2 + 9*s^4*t^4*r + 2*s^4*t^4 + 3*s^4*t^3*r^4 + 80*s^4*t^3*r^3 - 18*s^4*t^3*r^2 - 6*s^4*t^3*r + s^4*t^3 + 4*s^4*t^2*r^6 - 12*s^4*t^2*r^5 - 60*s^4*t^2*r^4 + 16*s^4*t^2*r^3 + 12*s^4*t^2*r^2 - 6*s^4*t^2*r + s^4*t^2 - 9*s^4*t*r^7 + 26*s^4*t*r^6 + 9*s^4*t*r^5 - 9*s^4*t*r^4 + s^4*t*r^3 + s^4*r^9 - 6*s^4*r^7 + 2*s^4*r^6 - 2*s^3*t^5 + 6*s^3*t^4*r + 4*s^3*t^4 + s^3*t^3*r^3 + 6*s^3*t^3*r^2 - 39*s^3*t^3*r + 12*s^3*t^3 - 18*s^3*t^2*r^4 + 7*s^3*t^2*r^3 + 60*s^3*t^2*r^2 - 33*s^3*t^2*r + 4*s^3*t^2 + 5*s^3*t*r^6 + 18*s^3*t*r^5 - 36*s^3*t*r^4 - 9*s^3*t*r^3 + 15*s^3*t*r^2 - 3*s^3*t*r - 6*s^3*r^7 + 4*s^3*r^6 + 9*s^3*r^5 - 6*s^3*r^4 + s^3*r^3 + 2*s^2*t^4 - 18*s^2*t^3*r + 10*s^2*t^3 + 5*s^2*t^2*r^3 + 42*s^2*t^2*r^2 - 45*s^2*t^2*r + 10*s^2*t^2 - 21*s^2*t*r^4 - 6*s^2*t*r^3 + 30*s^2*t*r^2 - 12*s^2*t*r + s^2*t + 2*s^2*r^6 + 9*s^2*r^5 - 12*s^2*r^4 + 3*s^2*r^3 + 3*s*t^3 - 15*s*t^2*r + 6*s*t^2 + 4*s*t*r^3 + 15*s*t*r^2 - 12*s*t*r + 2*s*t - 6*s*r^4 + 3*s*r^3 + t^2 - 3*t*r + t + r^3)/(s^2*t^2 - 3*s^2*t*r + s^2*t + s^2*r^3 + 2*s*t - 3*s*r + s + 1)*x^3 + (-3*s^5*t^7 + 6*s^5*t^6*r^2 + 18*s^5*t^6*r - 3*s^5*t^6 - 51*s^5*t^5*r^3 - 12*s^5*t^5*r^2 - 3*s^5*t^5*r + 3*s^5*t^5 + 12*s^5*t^4*r^5 + 129*s^5*t^4*r^4 - 72*s^5*t^4*r^3 + 57*s^5*t^4*r^2 - 24*s^5*t^4*r + 3*s^5*t^4 - 57*s^5*t^3*r^6 - 81*s^5*t^3*r^5 + 63*s^5*t^3*r^4 - 48*s^5*t^3*r^3 + 21*s^5*t^3*r^2 - 3*s^5*t^3*r + 6*s^5*t^2*r^8 + 60*s^5*t^2*r^7 - 6*s^5*t^2*r^6 + 6*s^5*t^2*r^5 - 3*s^5*t^2*r^4 - 9*s^5*t*r^9 - 15*s^5*t*r^8 + 3*s^5*t*r^7 + 3*s^5*r^10 - 12*s^4*t^6 + 18*s^4*t^5*r^2 + 66*s^4*t^5*r - 12*s^4*t^5 - 129*s^4*t^4*r^3 - 63*s^4*t^4*r^2 + 3*s^4*t^4*r + 9*s^4*t^4 + 30*s^4*t^3*r^5 + 228*s^4*t^3*r^4 - 27*s^4*t^3*r^3 + 33*s^4*t^3*r^2 - 27*s^4*t^3*r + 3*s^4*t^3 - 87*s^4*t^2*r^6 - 90*s^4*t^2*r^5 - 51*s^4*t^2*r^4 + 63*s^4*t^2*r^3 - 18*s^4*t^2*r^2 + 3*s^4*t^2*r + 6*s^4*t*r^8 + 39*s^4*t*r^7 + 57*s^4*t*r^6 - 33*s^4*t*r^5 + 3*s^4*t*r^4 - 18*s^4*r^8 + 6*s^4*r^7 - 18*s^3*t^5 + 18*s^3*t^4*r^2 + 90*s^3*t^4*r - 18*s^3*t^4 - 108*s^3*t^3*r^3 - 99*s^3*t^3*r^2 + 18*s^3*t^3*r + 9*s^3*t^3 + 24*s^3*t^2*r^5 + 117*s^3*t^2*r^4 + 90*s^3*t^2*r^3 - 51*s^3*t^2*r^2 - 30*s^3*t*r^6 - 54*s^3*t*r^5 - 45*s^3*t*r^4 + 48*s^3*t*r^3 - 9*s^3*t*r^2 + 6*s^3*r^7 + 27*s^3*r^6 - 18*s^3*r^5 + 3*s^3*r^4 - 12*s^2*t^4 + 6*s^2*t^3*r^2 + 54*s^2*t^3*r - 12*s^2*t^3 - 33*s^2*t^2*r^3 - 57*s^2*t^2*r^2 + 15*s^2*t^2*r + 3*s^2*t^2 + 6*s^2*t*r^5 + 21*s^2*t*r^4 + 45*s^2*t*r^3 - 27*s^2*t*r^2 + 3*s^2*t*r - 18*s^2*r^5 + 6*s^2*r^4 - 3*s*t^3 + 12*s*t^2*r - 3*s*t^2 - 3*s*t*r^3 - 9*s*t*r^2 + 3*s*t*r + 3*s*r^4)/(s^2*t^2 - 3*s^2*t*r + s^2*t + s^2*r^3 + 2*s*t - 3*s*r + s + 1)*x^2 + (3*s^5*t^7*r - 6*s^5*t^7 - 18*s^5*t^6*r^2 + 51*s^5*t^6*r - 12*s^5*t^6 + 3*s^5*t^5*r^4 + 12*s^5*t^5*r^3 - 129*s^5*t^5*r^2 + 57*s^5*t^5*r - 6*s^5*t^5 + 3*s^5*t^4*r^5 + 72*s^5*t^4*r^4 + 81*s^5*t^4*r^3 - 60*s^5*t^4*r^2 + 9*s^5*t^4*r - 3*s^5*t^3*r^7 - 57*s^5*t^3*r^6 - 63*s^5*t^3*r^5 + 6*s^5*t^3*r^4 + 15*s^5*t^3*r^3 - 3*s^5*t^3*r^2 + 24*s^5*t^2*r^8 + 48*s^5*t^2*r^7 - 6*s^5*t^2*r^6 - 3*s^5*t^2*r^5 - 3*s^5*t*r^10 - 21*s^5*t*r^9 + 3*s^5*t*r^8 + 3*s^5*r^11 + 12*s^4*t^6*r - 24*s^4*t^6 - 60*s^4*t^5*r^2 + 174*s^4*t^5*r - 42*s^4*t^5 + 15*s^4*t^4*r^4 - 3*s^4*t^4*r^3 - 321*s^4*t^4*r^2 + 141*s^4*t^4*r - 12*s^4*t^4 - 6*s^4*t^3*r^5 + 249*s^4*t^3*r^4 + 69*s^4*t^3*r^3 - 81*s^4*t^3*r^2 + 9*s^4*t^3*r - 3*s^4*t^2*r^7 - 102*s^4*t^2*r^6 - 159*s^4*t^2*r^5 + 99*s^4*t^2*r^4 - 18*s^4*t^2*r^3 + 3*s^4*t^2*r^2 + 21*s^4*t*r^8 + 87*s^4*t*r^7 - 39*s^4*t*r^6 + 3*s^4*t*r^5 - 18*s^4*r^9 + 6*s^4*r^8 + 18*s^3*t^5*r - 36*s^3*t^5 - 72*s^3*t^4*r^2 + 216*s^3*t^4*r - 54*s^3*t^4 + 24*s^3*t^3*r^4 - 57*s^3*t^3*r^3 - 252*s^3*t^3*r^2 + 111*s^3*t^3*r - 6*s^3*t^3 - 9*s^3*t^2*r^5 + 252*s^3*t^2*r^4 - 45*s^3*t^2*r^3 - 18*s^3*t^2*r^2 - 72*s^3*t*r^6 - 63*s^3*t*r^5 + 54*s^3*t*r^4 - 9*s^3*t*r^3 + 6*s^3*r^8 + 27*s^3*r^7 - 18*s^3*r^6 + 3*s^3*r^5 + 12*s^2*t^4*r - 24*s^2*t^4 - 36*s^2*t^3*r^2 + 114*s^2*t^3*r - 30*s^2*t^3 + 15*s^2*t^2*r^4 - 57*s^2*t^2*r^3 - 57*s^2*t^2*r^2 + 27*s^2*t^2*r + 3*s^2*t*r^5 + 75*s^2*t*r^4 - 33*s^2*t*r^3 + 3*s^2*t*r^2 - 18*s^2*r^6 + 6*s^2*r^5 + 3*s*t^3*r - 6*s*t^3 - 6*s*t^2*r^2 + 21*s*t^2*r - 6*s*t^2 + 3*s*t*r^4 - 15*s*t*r^3 + 3*s*t*r^2 + 3*s*r^5)/(s^2*t^2 - 3*s^2*t*r + s^2*t + s^2*r^3 + 2*s*t - 3*s*r + s + 1)*x + (s^5*t^8 - 9*s^5*t^7*r + s^5*t^7 + 2*s^5*t^6*r^3 + 33*s^5*t^6*r^2 - 6*s^5*t^6*r - s^5*t^6 - 15*s^5*t^5*r^4 - 64*s^5*t^5*r^3 + 21*s^5*t^5*r^2 + 3*s^5*t^5*r - s^5*t^5 + 2*s^5*t^4*r^6 + 48*s^5*t^4*r^5 + 54*s^5*t^4*r^4 - 40*s^5*t^4*r^3 + 6*s^5*t^4*r^2 - 15*s^5*t^3*r^7 - 64*s^5*t^3*r^6 + 21*s^5*t^3*r^5 + 3*s^5*t^3*r^4 - s^5*t^3*r^3 + 2*s^5*t^2*r^9 + 33*s^5*t^2*r^8 - 6*s^5*t^2*r^7 - s^5*t^2*r^6 - 9*s^5*t*r^10 + s^5*t*r^9 + s^5*r^12 + 5*s^4*t^7 - 39*s^4*t^6*r + 5*s^4*t^6 + 10*s^4*t^5*r^3 + 114*s^4*t^5*r^2 - 21*s^4*t^5*r - 4*s^4*t^5 - 57*s^4*t^4*r^4 - 161*s^4*t^4*r^3 + 51*s^4*t^4*r^2 + 9*s^4*t^4*r - 2*s^4*t^4 + 7*s^4*t^3*r^6 + 129*s^4*t^3*r^5 + 72*s^4*t^3*r^4 - 59*s^4*t^3*r^3 + 6*s^4*t^3*r^2 - 36*s^4*t^2*r^7 - 97*s^4*t^2*r^6 + 48*s^4*t^2*r^5 - 6*s^4*t^2*r^4 + s^4*t^2*r^3 + 4*s^4*t*r^9 + 39*s^4*t*r^8 - 15*s^4*t*r^7 + s^4*t*r^6 - 6*s^4*r^10 + 2*s^4*r^9 + 10*s^3*t^6 - 66*s^3*t^5*r + 10*s^3*t^5 + 19*s^3*t^4*r^3 + 144*s^3*t^4*r^2 - 27*s^3*t^4*r - 6*s^3*t^4 - 78*s^3*t^3*r^4 - 129*s^3*t^3*r^3 + 39*s^3*t^3*r^2 + 9*s^3*t^3*r - s^3*t^3 + 9*s^3*t^2*r^6 + 111*s^3*t^2*r^5 + 9*s^3*t^2*r^4 - 19*s^3*t^2*r^3 - 27*s^3*t*r^7 - 31*s^3*t*r^6 + 21*s^3*t*r^5 - 3*s^3*t*r^4 + 2*s^3*r^9 + 9*s^3*r^8 - 6*s^3*r^7 + s^3*r^6 + 10*s^2*t^5 - 54*s^2*t^4*r + 10*s^2*t^4 + 17*s^2*t^3*r^3 + 78*s^2*t^3*r^2 - 15*s^2*t^3*r - 4*s^2*t^3 - 45*s^2*t^2*r^4 - 31*s^2*t^2*r^3 + 9*s^2*t^2*r^2 + 3*s^2*t^2*r + 5*s^2*t*r^6 + 30*s^2*t*r^5 - 9*s^2*t*r^4 - 6*s^2*r^7 + 2*s^2*r^6 + 5*s*t^4 - 21*s*t^3*r + 5*s*t^3 + 7*s*t^2*r^3 + 15*s*t^2*r^2 - 3*s*t^2*r - s*t^2 - 9*s*t*r^4 + s*t*r^3 + s*r^6 + t^3 - 3*t^2*r + t^2 + t*r^3)/(s^2*t^2 - 3*s^2*t*r + s^2*t + s^2*r^3 + 2*s*t - 3*s*r + s + 1);
Ht2:=(-s^2*t^3 + s^2*t^2*r^2 + 2*s^2*t^2*r - 2*s^2*t*r^3 - s^2*t*r^2 + s^2*r^4 - 2*s*t^2 + 2*s*t*r^2 + 2*s*t*r - 2*s*r^3 - t + r^2)*x^2 + (s^2*t^3*r - 3*s^2*t^3 + s^2*t^2*r^2 + 6*s^2*t^2*r - s^2*t^2 - s^2*t*r^4 - 5*s^2*t*r^3 + 2*s^2*r^5 + 2*s*t^2*r - 4*s*t^2 + s*t*r^2 + 3*s*t*r - s*r^4 - s*r^3 + t*r - t)*x + s^2*t^4 - 4*s^2*t^3*r + s^2*t^2*r^3 + 6*s^2*t^2*r^2 - s^2*t^2*r - 4*s^2*t*r^4 + s^2*r^6 + 2*s*t^3 - 5*s*t^2*r + s*t*r^3 + 3*s*t*r^2 - s*r^4 + t^2 - t*r;
lt2:=(4*s^3*t^5 - 24*s^3*t^4*r + 8*s^3*t^4 + 8*s^3*t^3*r^3 + 36*s^3*t^3*r^2 - 24*s^3*t^3*r + 4*s^3*t^3 - 24*s^3*t^2*r^4 + 8*s^3*t^2*r^3 + 4*s^3*t*r^6 + 8*s^2*t^4 - 36*s^2*t^3*r + 12*s^2*t^3 + 12*s^2*t^2*r^3 + 24*s^2*t^2*r^2 - 12*s^2*t^2*r - 12*s^2*t*r^4 + 4*s^2*t*r^3 + 4*s*t^3 - 12*s*t^2*r + 4*s*t^2 + 4*s*t*r^3)/(s^4*t^6 - 6*s^4*t^5*r + 2*s^4*t^5 + 2*s^4*t^4*r^3 + 9*s^4*t^4*r^2 - 6*s^4*t^4*r + s^4*t^4 - 6*s^4*t^3*r^4 + 2*s^4*t^3*r^3 + s^4*t^2*r^6 + 4*s^3*t^5 - 18*s^3*t^4*r + 4*s^3*t^4 + 6*s^3*t^3*r^3 + 18*s^3*t^3*r^2 - 6*s^3*t^3*r - 12*s^3*t^2*r^4 + 2*s^3*t^2*r^3 + 2*s^3*t*r^6 + 6*s^2*t^4 - 18*s^2*t^3*r + 2*s^2*t^3 + 6*s^2*t^2*r^3 + 9*s^2*t^2*r^2 - 6*s^2*t*r^4 + s^2*r^6 + 4*s*t^3 - 6*s*t^2*r + 2*s*t*r^3 + t^2);
Gt2:=(-s^5*t^8 + 9*s^5*t^7*r - 2*s^5*t^7 - s^5*t^6*r^3 - 33*s^5*t^6*r^2 + 15*s^5*t^6*r - 2*s^5*t^6 + 6*s^5*t^5*r^4 + 64*s^5*t^5*r^3 - 48*s^5*t^5*r^2 + 15*s^5*t^5*r - 2*s^5*t^5 + s^5*t^4*r^6 - 21*s^5*t^4*r^5 - 54*s^5*t^4*r^4 + 64*s^5*t^4*r^3 - 33*s^5*t^4*r^2 + 9*s^5*t^4*r - s^5*t^4 - 3*s^5*t^3*r^7 + 40*s^5*t^3*r^6 - 21*s^5*t^3*r^5 + 6*s^5*t^3*r^4 - s^5*t^3*r^3 + s^5*t^2*r^9 - 6*s^5*t^2*r^8 - 3*s^5*t^2*r^7 + s^5*t^2*r^6 + s^5*t*r^9 - 5*s^4*t^7 + 39*s^4*t^6*r - 10*s^4*t^6 - 5*s^4*t^5*r^3 - 114*s^4*t^5*r^2 + 57*s^4*t^5*r - 7*s^4*t^5 + 21*s^4*t^4*r^4 + 161*s^4*t^4*r^3 - 129*s^4*t^4*r^2 + 36*s^4*t^4*r - 4*s^4*t^4 + 4*s^4*t^3*r^6 - 51*s^4*t^3*r^5 - 72*s^4*t^3*r^4 + 97*s^4*t^3*r^3 - 39*s^4*t^3*r^2 + 6*s^4*t^3*r - 9*s^4*t^2*r^7 + 59*s^4*t^2*r^6 - 48*s^4*t^2*r^5 + 15*s^4*t^2*r^4 - 2*s^4*t^2*r^3 + 2*s^4*t*r^9 - 6*s^4*t*r^8 + 6*s^4*t*r^7 - s^4*t*r^6 - s^4*r^9 - 10*s^3*t^6 + 66*s^3*t^5*r - 19*s^3*t^5 - 10*s^3*t^4*r^3 - 144*s^3*t^4*r^2 + 78*s^3*t^4*r - 9*s^3*t^4 + 27*s^3*t^3*r^4 + 129*s^3*t^3*r^3 - 111*s^3*t^3*r^2 + 27*s^3*t^3*r - 2*s^3*t^3 + 6*s^3*t^2*r^6 - 39*s^3*t^2*r^5 - 9*s^3*t^2*r^4 + 31*s^3*t^2*r^3 - 9*s^3*t^2*r^2 - 9*s^3*t*r^7 + 19*s^3*t*r^6 - 21*s^3*t*r^5 + 6*s^3*t*r^4 + s^3*r^9 + 3*s^3*r^7 - s^3*r^6 - 10*s^2*t^5 + 54*s^2*t^4*r - 17*s^2*t^4 - 10*s^2*t^3*r^3 - 78*s^2*t^3*r^2 + 45*s^2*t^3*r - 5*s^2*t^3 + 15*s^2*t^2*r^4 + 31*s^2*t^2*r^3 - 30*s^2*t^2*r^2 + 6*s^2*t^2*r + 4*s^2*t*r^6 - 9*s^2*t*r^5 + 9*s^2*t*r^4 - 2*s^2*t*r^3 - 3*s^2*r^7 - 5*s*t^4 + 21*s*t^3*r - 7*s*t^3 - 5*s*t^2*r^3 - 15*s*t^2*r^2 + 9*s*t^2*r - s*t^2 + 3*s*t*r^4 - s*t*r^3 + s*r^6 - t^3 + 3*t^2*r - t^2 - t*r^3)/(s^2*t^3 - 3*s^2*t^2*r + s^2*t^2 + s^2*t*r^3 + 2*s*t^2 - 3*s*t*r + s*r^3 + t)*x^3 + (-3*s^5*t^8 + 6*s^5*t^7*r^2 + 18*s^5*t^7*r - 3*s^5*t^7 - 51*s^5*t^6*r^3 - 12*s^5*t^6*r^2 - 3*s^5*t^6*r + 3*s^5*t^6 + 12*s^5*t^5*r^5 + 129*s^5*t^5*r^4 - 72*s^5*t^5*r^3 + 57*s^5*t^5*r^2 - 24*s^5*t^5*r + 3*s^5*t^5 - 57*s^5*t^4*r^6 - 81*s^5*t^4*r^5 + 63*s^5*t^4*r^4 - 48*s^5*t^4*r^3 + 21*s^5*t^4*r^2 - 3*s^5*t^4*r + 6*s^5*t^3*r^8 + 60*s^5*t^3*r^7 - 6*s^5*t^3*r^6 + 6*s^5*t^3*r^5 - 3*s^5*t^3*r^4 - 9*s^5*t^2*r^9 - 15*s^5*t^2*r^8 + 3*s^5*t^2*r^7 + 3*s^5*t*r^10 - 12*s^4*t^7 + 24*s^4*t^6*r^2 + 60*s^4*t^6*r - 15*s^4*t^6 - 174*s^4*t^5*r^3 + 3*s^4*t^5*r^2 + 6*s^4*t^5*r + 3*s^4*t^5 + 42*s^4*t^4*r^5 + 321*s^4*t^4*r^4 - 249*s^4*t^4*r^3 + 102*s^4*t^4*r^2 - 21*s^4*t^4*r - 141*s^4*t^3*r^6 - 69*s^4*t^3*r^5 + 159*s^4*t^3*r^4 - 87*s^4*t^3*r^3 + 18*s^4*t^3*r^2 + 12*s^4*t^2*r^8 + 81*s^4*t^2*r^7 - 99*s^4*t^2*r^6 + 39*s^4*t^2*r^5 - 6*s^4*t^2*r^4 - 9*s^4*t*r^9 + 18*s^4*t*r^8 - 3*s^4*t*r^7 - 3*s^4*r^10 - 18*s^3*t^6 + 36*s^3*t^5*r^2 + 72*s^3*t^5*r - 24*s^3*t^5 - 216*s^3*t^4*r^3 + 57*s^3*t^4*r^2 + 9*s^3*t^4*r + 54*s^3*t^3*r^5 + 252*s^3*t^3*r^4 - 252*s^3*t^3*r^3 + 72*s^3*t^3*r^2 - 6*s^3*t^3*r - 111*s^3*t^2*r^6 + 45*s^3*t^2*r^5 + 63*s^3*t^2*r^4 - 27*s^3*t^2*r^3 + 6*s^3*t*r^8 + 18*s^3*t*r^7 - 54*s^3*t*r^6 + 18*s^3*t*r^5 + 9*s^3*r^8 - 3*s^3*r^7 - 12*s^2*t^5 + 24*s^2*t^4*r^2 + 36*s^2*t^4*r - 15*s^2*t^4 - 114*s^2*t^3*r^3 + 57*s^2*t^3*r^2 - 3*s^2*t^3*r + 30*s^2*t^2*r^5 + 57*s^2*t^2*r^4 - 75*s^2*t^2*r^3 + 18*s^2*t^2*r^2 - 27*s^2*t*r^6 + 33*s^2*t*r^5 - 6*s^2*t*r^4 - 3*s^2*r^7 - 3*s*t^4 + 6*s*t^3*r^2 + 6*s*t^3*r - 3*s*t^3 - 21*s*t^2*r^3 + 15*s*t^2*r^2 - 3*s*t^2*r + 6*s*t*r^5 - 3*s*t*r^4)/(s^2*t^3 - 3*s^2*t^2*r + s^2*t^2 + s^2*t*r^3 + 2*s*t^2 - 3*s*t*r + s*r^3 + t)*x^2 + (3*s^5*t^8*r - 6*s^5*t^8 - 18*s^5*t^7*r^2 + 51*s^5*t^7*r - 12*s^5*t^7 + 3*s^5*t^6*r^4 + 12*s^5*t^6*r^3 - 129*s^5*t^6*r^2 + 57*s^5*t^6*r - 6*s^5*t^6 + 3*s^5*t^5*r^5 + 72*s^5*t^5*r^4 + 81*s^5*t^5*r^3 - 60*s^5*t^5*r^2 + 9*s^5*t^5*r - 3*s^5*t^4*r^7 - 57*s^5*t^4*r^6 - 63*s^5*t^4*r^5 + 6*s^5*t^4*r^4 + 15*s^5*t^4*r^3 - 3*s^5*t^4*r^2 + 24*s^5*t^3*r^8 + 48*s^5*t^3*r^7 - 6*s^5*t^3*r^6 - 3*s^5*t^3*r^5 - 3*s^5*t^2*r^10 - 21*s^5*t^2*r^9 + 3*s^5*t^2*r^8 + 3*s^5*t*r^11 + 12*s^4*t^7*r - 18*s^4*t^7 - 66*s^4*t^6*r^2 + 129*s^4*t^6*r - 30*s^4*t^6 + 12*s^4*t^5*r^4 + 63*s^4*t^5*r^3 - 228*s^4*t^5*r^2 + 87*s^4*t^5*r - 6*s^4*t^5 - 3*s^4*t^4*r^5 + 27*s^4*t^4*r^4 + 90*s^4*t^4*r^3 - 39*s^4*t^4*r^2 - 9*s^4*t^3*r^7 - 33*s^4*t^3*r^6 + 51*s^4*t^3*r^5 - 57*s^4*t^3*r^4 + 18*s^4*t^3*r^3 + 27*s^4*t^2*r^8 - 63*s^4*t^2*r^7 + 33*s^4*t^2*r^6 - 6*s^4*t^2*r^5 - 3*s^4*t*r^10 + 18*s^4*t*r^9 - 3*s^4*t*r^8 - 3*s^4*r^11 + 18*s^3*t^6*r - 18*s^3*t^6 - 90*s^3*t^5*r^2 + 108*s^3*t^5*r - 24*s^3*t^5 + 18*s^3*t^4*r^4 + 99*s^3*t^4*r^3 - 117*s^3*t^4*r^2 + 30*s^3*t^4*r - 18*s^3*t^3*r^5 - 90*s^3*t^3*r^4 + 54*s^3*t^3*r^3 - 6*s^3*t^3*r^2 - 9*s^3*t^2*r^7 + 51*s^3*t^2*r^6 + 45*s^3*t^2*r^5 - 27*s^3*t^2*r^4 - 48*s^3*t*r^7 + 18*s^3*t*r^6 + 9*s^3*r^9 - 3*s^3*r^8 + 12*s^2*t^5*r - 6*s^2*t^5 - 54*s^2*t^4*r^2 + 33*s^2*t^4*r - 6*s^2*t^4 + 12*s^2*t^3*r^4 + 57*s^2*t^3*r^3 - 21*s^2*t^3*r^2 - 15*s^2*t^2*r^5 - 45*s^2*t^2*r^4 + 18*s^2*t^2*r^3 - 3*s^2*t*r^7 + 27*s^2*t*r^6 - 6*s^2*t*r^5 - 3*s^2*r^8 + 3*s*t^4*r - 12*s*t^3*r^2 + 3*s*t^3*r + 3*s*t^2*r^4 + 9*s*t^2*r^3 - 3*s*t^2*r^2 - 3*s*t*r^5)/(s^2*t^3 - 3*s^2*t^2*r + s^2*t^2 + s^2*t*r^3 + 2*s*t^2 - 3*s*t*r + s*r^3 + t)*x + (s^5*t^9 - 9*s^5*t^8*r + s^5*t^8 + 2*s^5*t^7*r^3 + 33*s^5*t^7*r^2 - 6*s^5*t^7*r - s^5*t^7 - 15*s^5*t^6*r^4 - 64*s^5*t^6*r^3 + 21*s^5*t^6*r^2 + 3*s^5*t^6*r - s^5*t^6 + 2*s^5*t^5*r^6 + 48*s^5*t^5*r^5 + 54*s^5*t^5*r^4 - 40*s^5*t^5*r^3 + 6*s^5*t^5*r^2 - 15*s^5*t^4*r^7 - 64*s^5*t^4*r^6 + 21*s^5*t^4*r^5 + 3*s^5*t^4*r^4 - s^5*t^4*r^3 + 2*s^5*t^3*r^9 + 33*s^5*t^3*r^8 - 6*s^5*t^3*r^7 - s^5*t^3*r^6 - 9*s^5*t^2*r^10 + s^5*t^2*r^9 + s^5*t*r^12 + 3*s^4*t^8 - 21*s^4*t^7*r + 2*s^4*t^7 + 3*s^4*t^6*r^3 + 54*s^4*t^6*r^2 - 3*s^4*t^6*r - 4*s^4*t^6 - 9*s^4*t^5*r^4 - 80*s^4*t^5*r^3 + 12*s^4*t^5*r^2 + 9*s^4*t^5*r - s^4*t^5 - 2*s^4*t^4*r^6 + 18*s^4*t^4*r^5 + 60*s^4*t^4*r^4 - 26*s^4*t^4*r^3 + 6*s^4*t^3*r^7 - 16*s^4*t^3*r^6 - 9*s^4*t^3*r^5 + 6*s^4*t^3*r^4 - s^4*t^2*r^9 - 12*s^4*t^2*r^8 + 9*s^4*t^2*r^7 - 2*s^4*t^2*r^6 + 6*s^4*t*r^10 - s^4*t*r^9 - s^4*r^12 + 2*s^3*t^7 - 6*s^3*t^6*r - s^3*t^6 - 4*s^3*t^5*r^3 - 6*s^3*t^5*r^2 + 18*s^3*t^5*r - 5*s^3*t^5 + 39*s^3*t^4*r^4 - 7*s^3*t^4*r^3 - 18*s^3*t^4*r^2 + 6*s^3*t^4*r - 12*s^3*t^3*r^6 - 60*s^3*t^3*r^5 + 36*s^3*t^3*r^4 - 4*s^3*t^3*r^3 + 33*s^3*t^2*r^7 + 9*s^3*t^2*r^6 - 9*s^3*t^2*r^5 - 4*s^3*t*r^9 - 15*s^3*t*r^8 + 6*s^3*t*r^7 + 3*s^3*r^10 - s^3*r^9 - 2*s^2*t^6 + 18*s^2*t^5*r - 5*s^2*t^5 - 10*s^2*t^4*r^3 - 42*s^2*t^4*r^2 + 21*s^2*t^4*r - 2*s^2*t^4 + 45*s^2*t^3*r^4 + 6*s^2*t^3*r^3 - 9*s^2*t^3*r^2 - 10*s^2*t^2*r^6 - 30*s^2*t^2*r^5 + 12*s^2*t^2*r^4 + 12*s^2*t*r^7 - 3*s^2*t*r^6 - s^2*r^9 - 3*s*t^5 + 15*s*t^4*r - 4*s*t^4 - 6*s*t^3*r^3 - 15*s*t^3*r^2 + 6*s*t^3*r + 12*s*t^2*r^4 - 3*s*t^2*r^3 - 2*s*t*r^6 - t^4 + 3*t^3*r - t^3 - t^2*r^3)/(s^2*t^3 - 3*s^2*t^2*r + s^2*t^2 + s^2*t*r^3 + 2*s*t^2 - 3*s*t*r + s*r^3 + t);
Ht3:=(-s*t*r^2 + 2*s*t*r - s*t + s*r^4 - 2*s*r^3 + s*r^2)*x^2 + (s*t^3 - 6*s*t^2*r + 3*s*t^2 + 3*s*t*r^3 + 3*s*t*r^2 - 3*s*t*r - 3*s*r^4 + 2*s*r^3 + t^2 - 3*t*r + t + r^3)*x - s*t^2*r + s*t^2 + 2*s*t*r^3 - 2*s*t*r^2 - s*r^5 + s*r^4;
lt3:=(4*s^2*t^5 - 24*s^2*t^4*r + 8*s^2*t^4 + 8*s^2*t^3*r^3 + 36*s^2*t^3*r^2 - 24*s^2*t^3*r + 4*s^2*t^3 - 24*s^2*t^2*r^4 + 8*s^2*t^2*r^3 + 4*s^2*t*r^6 + 8*s*t^4 - 36*s*t^3*r + 12*s*t^3 + 12*s*t^2*r^3 + 24*s*t^2*r^2 - 12*s*t^2*r - 12*s*t*r^4 + 4*s*t*r^3 + 4*t^3 - 12*t^2*r + 4*t^2 + 4*t*r^3)/(t^4 - 6*t^3*r + 2*t^3 + 2*t^2*r^3 + 9*t^2*r^2 - 6*t^2*r + t^2 - 6*t*r^4 + 2*t*r^3 + r^6);
Gt3:=(-s^3*t^7 + 9*s^3*t^6*r - 2*s^3*t^6 - s^3*t^5*r^3 - 33*s^3*t^5*r^2 + 15*s^3*t^5*r - 2*s^3*t^5 + 6*s^3*t^4*r^4 + 64*s^3*t^4*r^3 - 48*s^3*t^4*r^2 + 15*s^3*t^4*r - 2*s^3*t^4 + s^3*t^3*r^6 - 21*s^3*t^3*r^5 - 54*s^3*t^3*r^4 + 64*s^3*t^3*r^3 - 33*s^3*t^3*r^2 + 9*s^3*t^3*r - s^3*t^3 - 3*s^3*t^2*r^7 + 40*s^3*t^2*r^6 - 21*s^3*t^2*r^5 + 6*s^3*t^2*r^4 - s^3*t^2*r^3 + s^3*t*r^9 - 6*s^3*t*r^8 - 3*s^3*t*r^7 + s^3*t*r^6 + s^3*r^9 - 3*s^2*t^6 + 24*s^2*t^5*r - 6*s^2*t^5 - 4*s^2*t^4*r^3 - 72*s^2*t^4*r^2 + 36*s^2*t^4*r - 5*s^2*t^4 + 21*s^2*t^3*r^4 + 97*s^2*t^3*r^3 - 81*s^2*t^3*r^2 + 27*s^2*t^3*r - 4*s^2*t^3 - 48*s^2*t^2*r^5 - 18*s^2*t^2*r^4 + 33*s^2*t^2*r^3 - 15*s^2*t^2*r^2 + 3*s^2*t^2*r + 6*s^2*t*r^7 + 19*s^2*t*r^6 - 9*s^2*t*r^5 + 3*s^2*t*r^4 - s^2*t*r^3 - s^2*r^9 - 3*s^2*r^7 + s^2*r^6 - 3*s*t^5 + 21*s*t^4*r - 6*s*t^4 - 5*s*t^3*r^3 - 48*s*t^3*r^2 + 27*s*t^3*r - 4*s*t^3 + 21*s*t^2*r^4 + 31*s*t^2*r^3 - 30*s*t^2*r^2 + 9*s*t^2*r - s*t^2 - 2*s*t*r^6 - 21*s*t*r^5 + 9*s*t*r^4 - s*t*r^3 + 3*s*r^7 - t^4 + 6*t^3*r - 2*t^3 - 2*t^2*r^3 - 9*t^2*r^2 + 6*t^2*r - t^2 + 6*t*r^4 - 2*t*r^3 - r^6)/(t^2 - 3*t*r + t + r^3)*x^3 + (-3*s^3*t^7 + 6*s^3*t^6*r^2 + 18*s^3*t^6*r - 3*s^3*t^6 - 51*s^3*t^5*r^3 - 12*s^3*t^5*r^2 - 3*s^3*t^5*r + 3*s^3*t^5 + 12*s^3*t^4*r^5 + 129*s^3*t^4*r^4 - 72*s^3*t^4*r^3 + 57*s^3*t^4*r^2 - 24*s^3*t^4*r + 3*s^3*t^4 - 57*s^3*t^3*r^6 - 81*s^3*t^3*r^5 + 63*s^3*t^3*r^4 - 48*s^3*t^3*r^3 + 21*s^3*t^3*r^2 - 3*s^3*t^3*r + 6*s^3*t^2*r^8 + 60*s^3*t^2*r^7 - 6*s^3*t^2*r^6 + 6*s^3*t^2*r^5 - 3*s^3*t^2*r^4 - 9*s^3*t*r^9 - 15*s^3*t*r^8 + 3*s^3*t*r^7 + 3*s^3*r^10 - 6*s^2*t^6 + 12*s^2*t^5*r^2 + 27*s^2*t^5*r - 3*s^2*t^5 - 81*s^2*t^4*r^3 + 6*s^2*t^4*r^2 - 24*s^2*t^4*r + 9*s^2*t^4 + 18*s^2*t^3*r^5 + 132*s^2*t^3*r^4 - 66*s^2*t^3*r^3 + 54*s^2*t^3*r^2 - 18*s^2*t^3*r - 45*s^2*t^2*r^6 - 66*s^2*t^2*r^5 + 36*s^2*t^2*r^4 - 24*s^2*t^2*r^3 + 9*s^2*t^2*r^2 + 36*s^2*t*r^7 + 3*s^2*t*r^5 - 3*s^2*t*r^4 - 9*s^2*r^8 + 3*s^2*r^7 - 3*s*t^5 + 6*s*t^4*r^2 + 9*s*t^4*r - 30*s*t^3*r^3 + 9*s*t^3*r^2 - 12*s*t^3*r + 3*s*t^3 + 6*s*t^2*r^5 + 30*s*t^2*r^4 - 15*s*t^2*r^3 + 12*s*t^2*r^2 - 3*s*t^2*r - 9*s*t*r^6 - 6*s*t*r^5 + 3*s*r^7)/(t^2 - 3*t*r + t + r^3)*x^2 + (3*s^3*t^7*r - 6*s^3*t^7 - 18*s^3*t^6*r^2 + 51*s^3*t^6*r - 12*s^3*t^6 + 3*s^3*t^5*r^4 + 12*s^3*t^5*r^3 - 129*s^3*t^5*r^2 + 57*s^3*t^5*r - 6*s^3*t^5 + 3*s^3*t^4*r^5 + 72*s^3*t^4*r^4 + 81*s^3*t^4*r^3 - 60*s^3*t^4*r^2 + 9*s^3*t^4*r - 3*s^3*t^3*r^7 - 57*s^3*t^3*r^6 - 63*s^3*t^3*r^5 + 6*s^3*t^3*r^4 + 15*s^3*t^3*r^3 - 3*s^3*t^3*r^2 + 24*s^3*t^2*r^8 + 48*s^3*t^2*r^7 - 6*s^3*t^2*r^6 - 3*s^3*t^2*r^5 - 3*s^3*t*r^10 - 21*s^3*t*r^9 + 3*s^3*t*r^8 + 3*s^3*r^11 + 6*s^2*t^6*r - 12*s^2*t^6 - 27*s^2*t^5*r^2 + 81*s^2*t^5*r - 18*s^2*t^5 + 3*s^2*t^4*r^4 - 6*s^2*t^4*r^3 - 132*s^2*t^4*r^2 + 45*s^2*t^4*r + 24*s^2*t^3*r^5 + 66*s^2*t^3*r^4 + 66*s^2*t^3*r^3 - 36*s^2*t^3*r^2 - 9*s^2*t^2*r^7 - 54*s^2*t^2*r^6 - 36*s^2*t^2*r^5 + 9*s^2*t^2*r^3 + 18*s^2*t*r^8 + 24*s^2*t*r^7 - 3*s^2*t*r^6 - 3*s^2*t*r^5 - 9*s^2*r^9 + 3*s^2*r^8 + 3*s*t^5*r - 6*s*t^5 - 9*s*t^4*r^2 + 30*s*t^4*r - 6*s*t^4 - 9*s*t^3*r^3 - 30*s*t^3*r^2 + 9*s*t^3*r + 12*s*t^2*r^5 + 15*s*t^2*r^4 + 6*s*t^2*r^3 - 3*s*t^2*r^2 - 3*s*t*r^7 - 12*s*t*r^6 + 3*s*r^8)/(t^2 - 3*t*r + t + r^3)*x + (s^3*t^8 - 9*s^3*t^7*r + s^3*t^7 + 2*s^3*t^6*r^3 + 33*s^3*t^6*r^2 - 6*s^3*t^6*r - s^3*t^6 - 15*s^3*t^5*r^4 - 64*s^3*t^5*r^3 + 21*s^3*t^5*r^2 + 3*s^3*t^5*r - s^3*t^5 + 2*s^3*t^4*r^6 + 48*s^3*t^4*r^5 + 54*s^3*t^4*r^4 - 40*s^3*t^4*r^3 + 6*s^3*t^4*r^2 - 15*s^3*t^3*r^7 - 64*s^3*t^3*r^6 + 21*s^3*t^3*r^5 + 3*s^3*t^3*r^4 - s^3*t^3*r^3 + 2*s^3*t^2*r^9 + 33*s^3*t^2*r^8 - 6*s^3*t^2*r^7 - s^3*t^2*r^6 - 9*s^3*t*r^10 + s^3*t*r^9 + s^3*r^12 + 3*s^2*t^7 - 24*s^2*t^6*r + 4*s^2*t^6 + 6*s^2*t^5*r^3 + 72*s^2*t^5*r^2 - 21*s^2*t^5*r - 36*s^2*t^4*r^4 - 97*s^2*t^4*r^3 + 48*s^2*t^4*r^2 - 6*s^2*t^4*r + s^2*t^4 + 5*s^2*t^3*r^6 + 81*s^2*t^3*r^5 + 18*s^2*t^3*r^4 - 19*s^2*t^3*r^3 - 27*s^2*t^2*r^7 - 33*s^2*t^2*r^6 + 9*s^2*t^2*r^5 + 3*s^2*t^2*r^4 + 4*s^2*t*r^9 + 15*s^2*t*r^8 - 3*s^2*t*r^7 - s^2*t*r^6 - 3*s^2*r^10 + s^2*r^9 + 3*s*t^6 - 21*s*t^5*r + 5*s*t^5 + 6*s*t^4*r^3 + 48*s*t^4*r^2 - 21*s*t^4*r + 2*s*t^4 - 27*s*t^3*r^4 - 31*s*t^3*r^3 + 21*s*t^3*r^2 - 3*s*t^3*r + 4*s*t^2*r^6 + 30*s*t^2*r^5 - 9*s*t^2*r^4 - 9*s*t*r^7 + s*t*r^6 + s*r^9 + t^5 - 6*t^4*r + 2*t^4 + 2*t^3*r^3 + 9*t^3*r^2 - 6*t^3*r + t^3 - 6*t^2*r^4 + 2*t^2*r^3 + t*r^6)/(t^2 - 3*t*r + t + r^3);
Ht4:=(-s*t*r + s*t + s*r^2 - s*r - r + 1)*x^2 + (-s*t^2 + s*t*r + s*t + s*r^3 - 2*s*r^2 - t + r)*x - s*t^2 + s*t*r^2 + s*t*r - s*r^3 - t + r^2;
lt4:=4*s^3*t^5 - 24*s^3*t^4*r + 8*s^3*t^4 + 8*s^3*t^3*r^3 + 36*s^3*t^3*r^2 - 24*s^3*t^3*r + 4*s^3*t^3 - 24*s^3*t^2*r^4 + 8*s^3*t^2*r^3 + 4*s^3*t*r^6 + 8*s^2*t^4 - 36*s^2*t^3*r + 12*s^2*t^3 + 12*s^2*t^2*r^3 + 24*s^2*t^2*r^2 - 12*s^2*t^2*r - 12*s^2*t*r^4 + 4*s^2*t*r^3 + 4*s*t^3 - 12*s*t^2*r + 4*s*t^2 + 4*s*t*r^3;
Gt4:=(-s^3*t^5 + 6*s^3*t^4*r - s^3*t^4 - 2*s^3*t^3*r^3 - 9*s^3*t^3*r^2 + s^3*t^3 + 6*s^3*t^2*r^4 + 9*s^3*t^2*r^2 - 6*s^3*t^2*r + s^3*t^2 - s^3*t*r^6 - 6*s^3*t*r^4 + 2*s^3*t*r^3 + s^3*r^6 - 3*s^2*t^4 + 15*s^2*t^3*r - 3*s^2*t^3 - 5*s^2*t^2*r^3 - 15*s^2*t^2*r^2 + 2*s^2*t^2 + 9*s^2*t*r^4 + 6*s^2*t*r^2 - 3*s^2*t*r - s^2*r^6 - 3*s^2*r^4 + s^2*r^3 - 3*s*t^3 + 12*s*t^2*r - 3*s*t^2 - 4*s*t*r^3 - 6*s*t*r^2 + s*t + 3*s*r^4 - t^2 + 3*t*r - t - r^3)*x^3 + (-3*s^3*t^5 + 21*s^3*t^4*r - 6*s^3*t^4 - 6*s^3*t^3*r^3 - 45*s^3*t^3*r^2 + 24*s^3*t^3*r - 3*s^3*t^3 + 24*s^3*t^2*r^4 + 21*s^3*t^2*r^3 - 18*s^3*t^2*r^2 + 3*s^3*t^2*r - 3*s^3*t*r^6 - 18*s^3*t*r^5 + 6*s^3*t*r^4 + 3*s^3*r^7 - 6*s^2*t^4 + 33*s^2*t^3*r - 9*s^2*t^3 - 9*s^2*t^2*r^3 - 45*s^2*t^2*r^2 + 18*s^2*t^2*r + 18*s^2*t*r^4 + 15*s^2*t*r^3 - 9*s^2*t*r^2 - 9*s^2*r^5 + 3*s^2*r^4 - 3*s*t^3 + 12*s*t^2*r - 3*s*t^2 - 3*s*t*r^3 - 9*s*t*r^2 + 3*s*t*r + 3*s*r^4)*x^2 + (-3*s^3*t^5*r + 21*s^3*t^4*r^2 - 6*s^3*t^4*r - 6*s^3*t^3*r^4 - 45*s^3*t^3*r^3 + 24*s^3*t^3*r^2 - 3*s^3*t^3*r + 24*s^3*t^2*r^5 + 21*s^3*t^2*r^4 - 18*s^3*t^2*r^3 + 3*s^3*t^2*r^2 - 3*s^3*t*r^7 - 18*s^3*t*r^6 + 6*s^3*t*r^5 + 3*s^3*r^8 - 6*s^2*t^4*r + 33*s^2*t^3*r^2 - 9*s^2*t^3*r - 9*s^2*t^2*r^4 - 45*s^2*t^2*r^3 + 18*s^2*t^2*r^2 + 18*s^2*t*r^5 + 15*s^2*t*r^4 - 9*s^2*t*r^3 - 9*s^2*r^6 + 3*s^2*r^5 - 3*s*t^3*r + 12*s*t^2*r^2 - 3*s*t^2*r - 3*s*t*r^4 - 9*s*t*r^3 + 3*s*t*r^2 + 3*s*r^5)*x - s^3*t^6 + 6*s^3*t^5*r - 2*s^3*t^5 - s^3*t^4*r^3 - 9*s^3*t^4*r^2 + 6*s^3*t^4*r - s^3*t^4 + s^3*t^2*r^6 + 9*s^3*t^2*r^5 - 6*s^3*t^2*r^4 + s^3*t^2*r^3 - 6*s^3*t*r^7 + 2*s^3*t*r^6 + s^3*r^9 - 3*s^2*t^5 + 15*s^2*t^4*r - 5*s^2*t^4 - 3*s^2*t^3*r^3 - 15*s^2*t^3*r^2 + 9*s^2*t^3*r - s^2*t^3 + 2*s^2*t*r^6 + 6*s^2*t*r^5 - 3*s^2*t*r^4 - 3*s^2*r^7 + s^2*r^6 - 3*s*t^4 + 12*s*t^3*r - 4*s*t^3 - 3*s*t^2*r^3 - 6*s*t^2*r^2 + 3*s*t^2*r + s*r^6 - t^3 + 3*t^2*r - t^2 - t*r^3;

//Equation K for the Kummer surfaces of the Jacobian of Crst.
//The equation is of the form:
//(k2^2-4*k1*k3)*k4^2+ PHI(k1,k2,k3)*k4+ PSI(k1,k2,k3)
//but written out in Horner form, which turns out to be a relatively compact way to express
//them.

K:=((-24*s+(-72*s^2+(-72*s^3-24*s^4*t)*t)*t)*t^4+((-24*s+(-24*s^2+(24*s^3+24*s^4*t)*t)*t)*t^3+((-48*s+(-96*s^2-48*s^3*t)*t)*t^3+((120*s^2+(192*s^3+72*s^4*t)*t)*t^3+((-72*s^2+(-96*s^3-72*s^4*t)*t)*t^2+((-96*s^2-48*s^3*t)*t^2+((168*s^3-72*s^4*t)*t^2+((-72*s^3+72*s^4*t)*t+(-48*s^3*t+(24*s^4*t-24*s^4*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*k1^4+(((8-8*s+((16-8*s)*s+(8*s^3+((-16+8*s)*s^3-8*s^4*t)*t)*t)*t)*t^3+((-96*s+(-264*s^2+(-240*s^3-72*s^4*t)*t)*t)*t^3+((144*s^2+(288*s^3+144*s^4*t)*t)*t^3+(((8-96*s)*s+((-24-128*s)*s^2+((-8-96*s)*s^3+24*s^4*t)*t)*t)*t^2+((-192*s^2+(48*s^3+144*s^4*t)*t)*t^2+((288*s^3-288*s^4*t)*t^2+(((-8-168*s)*s^2+((32+168*s)*s^3-24*s^4*t)*t)*t+((-96*s^3-72*s^4*t)*t+(144*s^4*t+((-8-80*s)*s^3+8*s^4*t)*r)*r)*r)*r)*r)*r)*r)*r)*r)*k1^3+(((-48*s+(-132*s^2+(-120*s^3-36*s^4*t)*t)*t)*t^3+(((-24+96*s)*s+((-72+192*s)*s^2+((-72+96*s)*s^3-24*s^4*t)*t)*t)*t^3+(((-24-60*s)*s+((-24-120*s)*s^2+((24-60*s)*s^3+24*s^4*t)*t)*t)*t^2+((-96*s^2+(24*s^3+72*s^4*t)*t)*t^2+(((-48+192*s)*s^2+(-192*s^4+48*s^4*t)*t)*t^2+(((-48-120*s)*s^2+((96+120*s)*s^3-48*s^4*t)*t)*t+((-48*s^3-36*s^4*t)*t+(((-24+96*s)*s^3-24*s^4*t)*t+((-24-60*s)*s^3+24*s^4*t)*r)*r)*r)*r)*r)*r)*r)*r)*k1^2+((((-24+24*s)*s+((-72+48*s)*s^2+((-72+24*s)*s^3-24*s^4*t)*t)*t)*t^3+(((-24-24*s)*s+((-24-48*s)*s^2+((24-24*s)*s^3+24*s^4*t)*t)*t)*t^2+(((-48+48*s)*s^2+(-48*s^4+48*s^4*t)*t)*t^2+(((-48-48*s)*s^2+((96+48*s)*s^3-48*s^4*t)*t)*t+(((-24+24*s)*s^3-24*s^4*t)*t+((-24-24*s)*s^3+24*s^4*t)*r)*r^2)*r)*r^2)*r)*k1+((-4+(-8-4*s)*s+((-16+(-8-8*s)*s)*s+((-24+(8-4*s)*s)*s^2+((-16+8*s)*s^3-4*s^4*t)*t)*t)*t)*t^2+(((-8+(-16-8*s)*s)*s+((-8+(32+8*s)*s)*s^2+((8-16*s)*s^3+8*s^4*t)*t)*t)*t+((-4+(-8-4*s)*s)*s^2+((-8+8*s)*s^3-4*s^4*t)*t)*r^3)*r^3)*k2)*k2)*k2)*k2+((((24-24*s)*s+(24*s^2+((-24+24*s)*s^3-24*s^4*t)*t)*t)*t^3+(((24-24*s)*s+((-312+48*s)*s^2+((-528-24*s)*s^3-192*s^4*t)*t)*t)*t^2+((-288*s^2+(504*s^3+648*s^4*t)*t)*t^2+(((-48+216*s)*s^2+(-696*s^4+48*s^4*t)*t)*t^2+((-264*s^3+((336+264*s)*s^3+168*s^4*t)*t)*t+((-288*s^3-648*s^4*t)*t+(((24+672*s)*s^3-24*s^4*t)*t+((-24-240*s)*s^3+24*s^4*t)*r)*r)*r)*r)*r)*r)*r)*r*k1^3+((((24-24*s)*s+((72-48*s)*s^2+((72-24*s)*s^3+24*s^4*t)*t)*t)*t^3+(((24+24*s)*s+((-264+48*s)*s^2+((-528+24*s)*s^3-240*s^4*t)*t)*t)*t^2+((-288*s^2+((-144+648*s)*s^2+((-288+792*s)*s^3-144*s^4*t)*t)*t)*t^2+(((-240+168*s)*s^2+(-888*s^4+240*s^4*t)*t)*t^2+(((-96-312*s)*s^2+((480+312*s)*s^3+120*s^4*t)*t)*t+((-288*s^3+((144-792*s)*s^3+144*s^4*t)*t)*t+(((24+912*s)*s^3-264*s^4*t)*t+((-120-336*s)*s^3+120*s^4*t)*r)*r)*r)*r)*r)*r)*r)*k1^2+(((8+(16+8*s)*s+((32+(16+16*s)*s)*s+((48+(-16+8*s)*s)*s^2+((32-16*s)*s^3+8*s^4*t)*t)*t)*t)*t^2+(((-144+144*s)*s^2+((-288+144*s)*s^3-144*s^4*t)*t)*t^3+((-288*s^2+(-288*s^4+288*s^4*t)*t)*t^2+(((16+(-112-128*s)*s)*s+((16+(224+128*s)*s)*s^2+((-16-112*s)*s^3-16*s^4*t)*t)*t)*t+(((144-144*s)*s^3+144*s^4*t)*t^2+((288*s^4-288*s^4*t)*t+((8+(-128-136*s)*s)*s^2+((16+128*s)*s^3+8*s^4*t)*t)*r)*r)*r)*r)*r)*r)*k1+(((-24+(-48-24*s)*s)*s+((-72-24*s^2)*s^2+((-72+48*s)*s^3-24*s^4*t)*t)*t)*t^2+(((-24+(-48-24*s)*s)*s+((-24+(96+24*s)*s)*s^2+((24-48*s)*s^3+24*s^4*t)*t)*t)*t+(((24+(48+24*s)*s)*s^2+((48-48*s)*s^3+24*s^4*t)*t)*t+((-24+(-48-24*s)*s)*s^2+((-48+48*s)*s^3-24*s^4*t)*t)*r)*r^2)*r)*r*k2)*k2)*k2+(((-16*s+((-16+(-288-16*s)*s)*s+((-32-488*s)*s^2+(-16-216*s)*s^3*t)*t)*t)*t^2+(((-96-192*s)*s+((-144+552*s)*s^2+(-48+648*s)*s^3*t)*t)*t^2+((72*s^3-648*s^4*t)*t^2+(((-16+(-160-216*s)*s)*s+((-288+(736+216*s)*s)*s^2+((-488+432*s)*s^3-216*s^4*t)*t)*t)*t+(((-192-480*s)*s^2+((552-1296*s)*s^3+648*s^4*t)*t)*t+(((72+1296*s)*s^3-648*s^4*t)*t+((-216-432*s)*s^3+(-16+216*s)*s^3*t)*r)*r)*r)*r)*r)*r)*k1^2+((((-144+144*s)*s^2+((-288+144*s)*s^3-144*s^4*t)*t)*t^3+(((24+(-240+24*s)*s)*s+((72-264*s^2)*s^2+((72+240*s)*s^3+24*s^4*t)*t)*t)*t^2+(((24+(-96-120*s)*s)*s+((-264+(480+120*s)*s)*s^2+((-528+120*s)*s^3-240*s^4*t)*t)*t)*t+(((-288-288*s)*s^2+((648-792*s)*s^3+792*s^4*t)*t)*t+(((-24+(168+912*s)*s)*s^2+((-48-888*s)*s^3-24*s^4*t)*t)*t+((24+(-312-336*s)*s)*s^2+((48+312*s)*s^3+24*s^4*t)*t)*r)*r)*r)*r)*r)*k1+(((-24+(-48-24*s)*s)*s+((-72-24*s^2)*s^2+((-72+48*s)*s^3-24*s^4*t)*t)*t)*t^2+(((-24+(-48-24*s)*s)*s+((-24+(96+24*s)*s)*s^2+((24-48*s)*s^3+24*s^4*t)*t)*t)*t+(((-48+(-96-48*s)*s)*s+((-132+(24-36*s)*s)*s^2+((-120+72*s)*s^3-36*s^4*t)*t)*t)*t+(((96+(192+96*s)*s)*s^2+((192-192*s)*s^3+96*s^4*t)*t)*t+((-60+(-120-60*s)*s)*s^2+((-120+120*s)*s^3-60*s^4*t)*t)*r)*r)*r)*r)*k2)*k2+((((24+(-48+24*s)*s)*s+((24-24*s^2)*s^2+((-24+48*s)*s^3-24*s^4*t)*t)*t)*t^2+(((24-24*s^2)*s+((-312+(336+24*s)*s)*s^2+((-528+168*s)*s^3-192*s^4*t)*t)*t)*t+(((-288-288*s)*s^2+((504-648*s)*s^3+648*s^4*t)*t)*t+(((-24+(216+672*s)*s)*s^2+(-696*s^4+24*s^4*t)*t)*t+((-24+(-264-240*s)*s)*s^2+((48+264*s)*s^3-24*s^4*t)*t)*r)*r)*r)*r)*k1+((8+(8+(-8-8*s)*s)*s+((16+(-24+(32+8*s)*s)*s)*s+((-8-24*s)*s^3+((-16+24*s)*s^3-8*s^4*t)*t)*t)*t)*t+(((-96+(-192-96*s)*s)*s+((-264+(48-72*s)*s)*s^2+((-240+144*s)*s^3-72*s^4*t)*t)*t)*t+(((144+(288+144*s)*s)*s^2+((288-288*s)*s^3+144*s^4*t)*t)*t+((-8+(-96+(-168-80*s)*s)*s)*s+((-8+(-128+168*s)*s)*s^2+((8-96*s)*s^3+8*s^4*t)*t)*t)*r)*r)*r)*k2+(((-48+(-96-48*s)*s)*s+((-96-48*s)*s^2-48*s^3*t)*t)*t+(((-24+(120+(168+24*s)*s)*s)*s+((-72+(192-72*s)*s)*s^2+((-72+72*s)*s^3-24*s^4*t)*t)*t)*t+((-24+(-72+(-72-24*s)*s)*s)*s+((-24+(-96+72*s)*s)*s^2+((24-72*s)*s^3+24*s^4*t)*t)*t)*r)*r)*k3)*k3)*k3)*k3+(((-4+(-8*s-4*s^2*t)*t)*t^2+((-8*s+8*s^2*t)*t-4*r^3*s^2)*r^3)*k1^3+((-12*s-12*s^2*t)*t^2+((-12*s+12*s^2*t)*t+(12*s^2*t-12*s^2*r)*r^2)*r)*r*k1^2*k2+(((-24*s-24*s^2*t)*t^2+((-24*s+24*s^2*t)*t+((-48*s-36*s^2*t)*t+(96*s^2*t-60*s^2*r)*r)*r)*r)*k1^2+((-4*s+4+(4*s^2-4*s^2*t)*t)*t+((-48*s-36*s^2*t)*t+(72*s^2*t+((-4-40*s)*s+4*s^2*t)*r)*r)*r)*k1*k2+(((-48*s-36*s^2*t)*t+(((-24+96*s)*s-24*s^2*t)*t+((-24-60*s)*s+24*s^2*t)*r)*r)*k1+(((-12+12*s)*s-12*s^2*t)*t+((-12-12*s)*s+12*s^2*t)*r)*k2+(-4+(-8-4*s)*s+((-8+8*s)*s-4*s^2*t)*t)*k3)*k3)*k3+(-4*k1*k3+k2^2)*k4)*k4;

//Cubic forms in k1,k2,k3,k4 such that
//(k1,k2,k3,k4) :-> (xi1,xi2,xi3,xi4)
//defines a rational map from the Kummer surface of Crst to the Kummer surface 
//of Ct_rst, induced by the (3,3)-isogenies between the Jacobians.
//The forms are written in Horner form. These expressions are by far the most
//voluminous in this file.

xi1:=(((-16+(44-16*s)*s)*s+((-76+(180-44*s)*s)*s^2+((-144+(276-40*s)*s)*s^3+((-136+(188-12*s)*s)*s^4+((-64+48*s)*s^5-12*s^6*t)*t)*t)*t)*t)*t^5+(((16+32*s)*s+((116+(-44+40*s)*s)*s^2+((276+(-352+64*s)*s)*s^3+((292+(-444+24*s)*s)*s^4+((140-168*s)*s^5+24*s^6*t)*t)*t)*t)*t)*t^4+(((-16-12*s)*s+((16+(-116+(-60-12*s)*s)*s)*s+((152+(-288+(204-40*s)*s)*s)*s^2+((432+(-268+(588-12*s)*s)*s)*s^3+((544+(-56+336*s)*s)*s^4+((320+24*s)*s^5+72*s^6*t)*t)*t)*t)*t)*t)*t^3+(((16+(92+36*s)*s)*s+((-204+(32+80*s)*s)*s^2+((-1292+(-676-348*s)*s)*s^3+((-2412+(-1096-408*s)*s)*s^4+((-1844-480*s)*s^5-504*s^6*t)*t)*t)*t)*t)*t^3+(((-16+(-216-216*s)*s)*s+((668+(308+64*s)*s)*s^3+((80+(3768+(1696+240*s)*s)*s)*s^3+((96+(4540+1140*s)*s)*s^4+(32+1656*s)*s^5*t)*t)*t)*t)*t^3+(((28+36*s)*s^2+((80+(724+(16+48*s)*s)*s)*s^2+((-16+(-972+(-956-48*s)*s)*s)*s^3+((-224+(-4236-1128*s)*s)*s^4+(-128-2520*s)*s^5*t)*t)*t)*t)*t^2+(((96+(-116-144*s)*s)*s^2+((208+(-484+96*s)*s)*s^3+((464+(1720+636*s)*s)*s^4+((256+2040*s)*s^5-48*s^6*t)*t)*t)*t)*t^2+(((-16+(-680+120*s)*s)*s^2+((16+(-1844+(-412-264*s)*s)*s)*s^3+((64+(-1916-1272*s)*s)*s^4+(32-480*s)*s^5*t)*t)*t)*t^2+(((104+108*s)*s^3+((16+(1688+(412+60*s)*s)*s)*s^3+((-240+(2864+744*s)*s)*s^4+(-208+1500*s)*s^5*t)*t)*t)*t+(((80+(-388-180*s)*s)*s^3+((404+(-1776-216*s)*s)*s^4+((1004-1320*s)*s^5+360*s^6*t)*t)*t)*t+(((-16+(-400+368*s)*s)*s^3+((32+(-1172+300*s)*s)*s^4+(32-1008*s)*s^5*t)*t)*t+((60+60*s)*s^4+((32+(620+48*s)*s)*s^4+((-152+936*s)*s^5-72*s^6*t)*t)*t+(-180*s^5+((104-288*s)*s^5+204*s^6*t)*t+(-192*s^6*t+60*s^6*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*k1^3+((((8-8*s)*s+((-16+(112+(-104+16*s)*s)*s)*s+((-80+(368+(-264+32*s)*s)*s)*s^2+((-160+(512+(-248+16*s)*s)*s)*s^3+((-160+(328-80*s)*s)*s^4+((-80+80*s)*s^5-16*s^6*t)*t)*t)*t)*t)*t)*t^4+(((-16-16*s)*s+((8+(-8-16*s^2)*s)*s+((120+(-32+(296-32*s)*s)*s)*s^2+((400+(-208+(528-16*s)*s)*s)*s^3+((560+(-272+248*s)*s)*s^4+((360-104*s)*s^5+88*s^6*t)*t)*t)*t)*t)*t)*t^3+(((24+(120+48*s)*s)*s+((24+(-168+(48-24*s)*s)*s)*s+((120+(-1272+(-528-504*s)*s)*s)*s^2+((240+(-2568+(-720-432*s)*s)*s)*s^3+((240+(-2112-264*s)*s)*s^4+((120-624*s)*s^5+24*s^6*t)*t)*t)*t)*t)*t)*t^3+(((-40+(-296-264*s)*s)*s+((-248+(224+(312+152*s)*s)*s)*s^2+((-576+(3120+(1632+328*s)*s)*s)*s^3+((-640+(4384+1056*s)*s)*s^4+((-344+1784*s)*s^5-72*s^6*t)*t)*t)*t)*t)*t^3+(((8+(56+48*s)*s)*s+((288+(1008+(248+64*s)*s)*s)*s^2+((968+(-528+(-776-64*s)*s)*s)*s^3+((1368+(-4272-1152*s)*s)*s^4+((944-2792*s)*s^5+264*s^6*t)*t)*t)*t)*t)*t^2+(((-312-192*s)*s^3+((48+(-456+(-1608-144*s)*s)*s)*s^2+((144+(-672+(1008+648*s)*s)*s)*s^3+((144+(-360+2304*s)*s)*s^4+(48-144*s)*s^5*t)*t)*t)*t)*t^2+(((-56+(-448+360*s)*s)*s^2+((-272+(-632+(304-344*s)*s)*s)*s^3+((-512+(-1144-1440*s)*s)*s^4+((-432-672*s)*s^5-136*s^6*t)*t)*t)*t)*t^2+(((16+(160+144*s)*s)*s^2+((344+(2128+(528+80*s)*s)*s)*s^3+((328+(3128+912*s)*s)*s^4+((88+1944*s)*s^5+88*s^6*t)*t)*t)*t)*t+(((-48+(-672-240*s)*s)*s^3+((-48+(-744+(-2880-216*s)*s)*s)*s^3+((-96+(-24-1800*s)*s)*s^4+(-48+288*s)*s^5*t)*t)*t)*t+(((-16+(-144+616*s)*s)*s^3+((336+(-160+464*s)*s)*s^4+((544-872*s)*s^5+192*s^6*t)*t)*t)*t+((8+(88+80*s)*s)*s^3+((56+(600-8*s)*s)*s^4+((-544+872*s)*s^5-432*s^6*t)*t)*t+((-24-240*s)*s^4+((-24+(96-240*s)*s)*s^4+((-48+480*s)*s^5-24*s^6*t)*t)*t+(((32-320*s)*s^5+32*s^6*t)*t+((8+80*s)*s^5-8*s^6*t)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*k1^2+(((-4+(-8-4*s)*s+(-4+(-12+(-40+(-4-4*s)*s)*s)*s+((-24+(40+(-136+(52-8*s)*s)*s)*s)*s+((-60+(200+(-248+(100-4*s)*s)*s)*s)*s^2+((-80+(300+(-208+48*s)*s)*s)*s^3+((-60+(196-64*s)*s)*s^4+((-24+48*s)*s^5-4*s^6*t)*t)*t)*t)*t)*t)*t)*t^3+((12+(36+(36+12*s)*s)*s+((84+(168+(84-48*s)*s)*s)*s+((240+(192+(108-180*s)*s)*s)*s^2+((360+(-72+(108-120*s)*s)*s)*s^3+((300+(-228+48*s)*s)*s^4+((132-96*s)*s^5+24*s^6*t)*t)*t)*t)*t)*t)*t^3+(((-96+(-120-48*s)*s)*s+((-588+(-288+(180+96*s)*s)*s)*s^2+((-1440+(72+(408+144*s)*s)*s)*s^3+((-1752+(528+180*s)*s)*s^4+((-1056+288*s)*s^5-252*s^6*t)*t)*t)*t)*t)*t^3+((-4+(4+(44+36*s)*s)*s+((-12+(232+(380+(216+40*s)*s)*s)*s)*s+((16+(1152+(36+(-220-40*s)*s)*s)*s)*s^2+((104+(2184+(-956-416*s)*s)*s)*s^3+((156+(1820-656*s)*s)*s^4+((100+560*s)*s^5+24*s^6*t)*t)*t)*t)*t)*t)*t^2+(((36+(-72+(-300-120*s)*s)*s)*s+((192+(-504+(-1272-264*s)*s)*s)*s^2+((360+(-1200+(-108+240*s)*s)*s)*s^3+((336+(-1272+864*s)*s)*s^4+((180-504*s)*s^5+48*s^6*t)*t)*t)*t)*t)*t^2+(((-96+(144+336*s)*s)*s^2+((-432+(840+(696-144*s)*s)*s)*s^3+((-624+(576-504*s)*s)*s^4+((-432-72*s)*s^5-144*s^6*t)*t)*t)*t)*t^2+(((-8+(20+(112+84*s)*s)*s)*s+((-40+(308+(916+(332+44*s)*s)*s)*s)*s^2+((-64+(492+(832+400*s)*s)*s)*s^3+((-80+(220+540*s)*s)*s^4+((-88+64*s)*s^5-40*s^6*t)*t)*t)*t)*t)*t+(((36+(-156+(-468-132*s)*s)*s)*s^2+((36+(-1152+(-1836-120*s)*s)*s)*s^3+((-156+(-756-864*s)*s)*s^4+((-180+144*s)*s^5-24*s^6*t)*t)*t)*t)*t+(((-48+(216+384*s)*s)*s^3+((372+(960+324*s)*s)*s^4+(792*s^5+324*s^6*t)*t)*t)*t+((-4+(12+(60+44*s)*s)*s)*s^2+((-4+(48+(188-32*s)*s)*s)*s^3+((28+(-700+136*s)*s)*s^4+((52-656*s)*s^5+24*s^6*t)*t)*t)*t+((12+(-48-132*s)*s)*s^3+((-24-48*s^2)*s^4+((-84+456*s)*s^5-48*s^6*t)*t)*t+(((72-216*s)*s^5+72*s^6*t)*t+(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120+(-69480-19872*s)*s)*s)*s^5+((-31320+(180432-30816*s)*s)*s^6+((-18000+67680*s)*s^7-4176*s^8*t)*t)*t)*t)*t)*t)*t+((-324+(-648-324*s)*s)*s^3+((1332+(-468+(1908-2016*s)*s)*s)*s^3+((11916+(-7956+(17784-4248*s)*s)*s)*s^4+((40536+(-51768+(19260-2736*s)*s)*s)*s^5+((65304+(-96336-4680*s)*s)*s^6+((50004-53568*s)*s^7+14652*s^8*t)*t)*t)*t)*t)*t+((972+972*s)*s^4+((72+(-2232+(7380+7632*s)*s)*s)*s^3+((504+(-18828+(21204+15480*s)*s)*s)*s^4+((1368+(-52704+(34812+8280*s)*s)*s)*s^5+((1800+(-59868+26208*s)*s)*s^6+((1152-23760*s)*s^7+288*s^8*t)*t)*t)*t)*t)*t+((108+(-756+108*s)*s)*s^4+((-504+(2268+(-7452+288*s)*s)*s)*s^4+((-2916+(16308+(-13680+252*s)*s)*s)*s^5+((-6120+(34272-4320*s)*s)*s^6+((-5508+20880*s)*s^7-1800*s^8*t)*t)*t)*t)*t+((-648-648*s)*s^5+((1260+(-3384-2196*s)*s)*s^5+((5688+(-9864-1728*s)*s)*s^6+((8172-10656*s)*s^7+3744*s^8*t)*t)*t)*t+(972*s^6+((36+(-1476+3492*s)*s)*s^5+((144+(-4968+2484*s)*s)*s^6+((180-3744*s)*s^7+72*s^8*t)*t)*t)*t+((108+108*s)*s^6+((-144+(1044+144*s)*s)*s^6+((-360+2016*s)*s^7-216*s^8*t)*t)*t+(-324*s^7+((216-504*s)*s^7+252*s^8*t)*t+(-144*s^8*t+36*s^8*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*k3+((1+(3+(9+3*s)*s+(3+(24+(39+(18+3*s)*s)*s)*s+(1+(21+(81+(93+(42+(9+s)*s)*s)*s)*s)*s+((6+(60+(144+(120+(42+6*s)*s)*s)*s)*s)*s+((15+(90+(141+(78+15*s)*s)*s)*s)*s^2+((20+(75+(72+20*s)*s)*s)*s^3+((15+(33+15*s)*s)*s^4+((6+6*s)*s^5+s^6*t)*t)*t)*t)*t)*t)*t)*t)*t)*t^3+((-9*s-9+(-18+(-99+(-90-18*s)*s)*s+(-9+(-153+(-441+(-351+(-99-9*s)*s)*s)*s)*s+((-63+(-522+(-999+(-648+(-171-18*s)*s)*s)*s)*s)*s+((-180+(-918+(-1206+(-558-90*s)*s)*s)*s)*s^2+((-270+(-882+(-738-180*s)*s)*s)*s^3+((-225+(-441-180*s)*s)*s^4+((-99-90*s)*s^5-18*s^6*t)*t)*t)*t)*t)*t)*t)*t)*t^3+((27+(72+27*s)*s+(27+(333+(621+(279+27*s)*s)*s)*s+((234+(1503+(2106+(918+126*s)*s)*s)*s)*s+((801+(3321+(3447+(1206+135*s)*s)*s)*s)*s^2+((1404+(3870+(2700+540*s)*s)*s)*s^3+((1341+(2286+810*s)*s)*s^4+((666+540*s)*s^5+135*s^6*t)*t)*t)*t)*t)*t)*t)*t^3+((3*s+3+(-21+(-156+(-159-21*s)*s)*s+(3+(-300+(-1419+(-1260+(-237+3*s)*s)*s)*s)*s+((21+(-1419+(-4662+(-3402+(-645+6*s)*s)*s)*s)*s)*s+((60+(-3177+(-7266+(-3864-510*s)*s)*s)*s)*s^2+((90+(-3702+(-5424-1560*s)*s)*s)*s^3+((75+(-2175-1560*s)*s)*s^4+((33-510*s)*s^5+6*s^6*t)*t)*t)*t)*t)*t)*t)*t)*t^2+((-18+(-45-18*s)*s+(-18+(-54+(36+(-18-18*s)*s)*s)*s+((-153+(351+(1395+(405-81*s)*s)*s)*s)*s+((-522+(1944+(4203+(1152-90*s)*s)*s)*s)*s^2+((-918+(3564+(4779+855*s)*s)*s)*s^3+((-882+(2862+1890*s)*s)*s^4+((-441+855*s)*s^5-90*s^6*t)*t)*t)*t)*t)*t)*t)*t^2+((27+(171+(171+27*s)*s)*s+((333+(1116+(945+252*s)*s)*s)*s+((1503+(2745+(1800+666*s)*s)*s)*s^2+((3321+(3069+(1269+540*s)*s)*s)*s^3+((3870+(1431+162*s)*s)*s^4+((2286+162*s)*s^5+540*s^6*t)*t)*t)*t)*t)*t)*t^2+((3+(6+3*s)*s+(3+(-156+(-429+(-162+3*s)*s)*s)*s+((24+(-1419+(-2820+(-1098+12*s)*s)*s)*s)*s+((81+(-4662+(-6615+(-2304+15*s)*s)*s)*s)*s^2+((144+(-7266+(-6678-1560*s)*s)*s)*s^3+((141+(-5424-2421*s)*s)*s^4+((72-1560*s)*s^5+15*s^6*t)*t)*t)*t)*t)*t)*t)*t+((-9+(-45+(-45-9*s)*s)*s+((-99+(36+(81-72*s)*s)*s)*s+((-441+(1395+(1314-198*s)*s)*s)*s^2+((-999+(4203+(2889-180*s)*s)*s)*s^3+((-1206+(4779+1890*s)*s)*s^4+((-738+1890*s)*s^5-180*s^6*t)*t)*t)*t)*t)*t)*t+(((72+(171+72*s)*s)*s+((621+(945+477*s)*s)*s^2+((2106+(1800+1080*s)*s)*s^3+((3447+(1269+810*s)*s)*s^4+((2700+162*s)*s^5+810*s^6*t)*t)*t)*t)*t)*t+(1+(3+(3+s)*s)*s+((9+(-159+(-162+6*s)*s)*s)*s+((39+(-1260+(-1098+18*s)*s)*s)*s^2+((93+(-3402+(-2304+20*s)*s)*s)*s^3+((120+(-3864-1560*s)*s)*s^4+((78-1560*s)*s^5+20*s^6*t)*t)*t)*t)*t)*t+((-9+(-18-9*s)*s)*s+((-90+(-18-72*s)*s)*s^2+((-351+(405-198*s)*s)*s^3+((-648+(1152-180*s)*s)*s^4+((-558+855*s)*s^5-180*s^6*t)*t)*t)*t)*t+((27+27*s)*s^2+((279+252*s)*s^3+((918+666*s)*s^4+((1206+540*s)*s^5+540*s^6*t)*t)*t)*t+((3+(-21+3*s)*s)*s^2+((18+(-237+12*s)*s)*s^3+((42+(-645+15*s)*s)*s^4+((42-510*s)*s^5+15*s^6*t)*t)*t)*t+((-18-18*s)*s^3+((-99-81*s)*s^4+((-171-90*s)*s^5-90*s^6*t)*t)*t+(27*s^4+(126*s^5+135*s^6*t)*t+((3*s+3)*s^4+((9+6*s)*s^5+6*s^6*t)*t+(-9*s^5-18*s^6*t+s^6*r^2)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*r)*k4)*k4)*k4;

//Automorphisms of level structure. Viewing k(r,s,t) as the function field
//of the moduli space parametrizing (3,3) level structures, there is an action of
//PGL(2,3) on it. The following involutions generate it.
psi1:=[ t/r^2, r^3*s/t, t^2/r^3];
psi2:=[ r, 1/(s*(r^3 - 3*r*t + t^2 + t)), t];
psi3:=[ r,t*(s*t+1)/r^3,r^3*s/(s*t + 1)];

//In addition there is an involution coming from the fact that a (3,3)-level structure
//induces a (3,3)-level structure on the -3 twist of the isogenous Abelian variety.
//that involution is psi0'=psi3*psi1*psi2*psi3*psi0, where psi0 is the involution given below.
psi0:=[-s*(r-1)*(r^2-t)*(delta5-r)/(r*s - s*t - 1)^2/delta4,(r*s - s*t -1)^3*delta4^2/s/t/(r-1)^3/Delta,s^2*(r-1)^3*(r^2-t)^3/(r*s-s*t-1)^3/delta4^2];
