k:=Integers();
Py<y0,y1,y2,y3,y4>:=PolynomialRing(k,5);
Pt<t1,t2,t3>:=PolynomialRing(k,3);
Ka<a1,a2,a3,a4>:=FunctionField(k,4);
KaX<X>:=PolynomialRing(Ka);
//the defining homogeneous equation of the model of the Burkhardt Quartic we use:
B:=y0^4+y0*y1^3+y0*y2^3+y0*y3^3+3*y1*y2*y3*y4+y0*y4^3;

//A description of a birational map A^3->B given by (y0:y1:y2:y3:y4)
//as polynomials in (t1,t2,t3)
phi:=[t1^3-3*t1^2*t3-3*t1*t2^2-3*t1*t2*t3-t2^3-1,
-t1^3+3*t1^2*t3-3*t1*t3^2+t2^3+1,
-t1^4+t1^3*t2+3*t1^3*t3-3*t1^2*t2*t3-3*t1^2*t3^2-2*t1*t2^3-3*t1*t2^2*t3+t1-t2^4-t2,
-t1^4+4*t1^3*t3+3*t1^2*t2^2+3*t1^2*t2*t3-3*t1^2*t3^2+t1*t2^3-3*t1*t2^2*t3-3*t1*t2*t3^2+t1-t2^3*t3-t3,
-t1^4-t1^3*t2+2*t1^3*t3+3*t1^2*t2*t3+t1*t2^3+3*t1*t2^2*t3+t1+t2^4+t2^3*t3+t2+t3];

//A list of descriptions of a birational map B->P^3
//given as a list of lists (t0:t1:t2:t3) as homogeneous polynomials in
//(y0:y1:y2:y3:y4)
psis:=[[y0^3-y0^2*y1+y0*y1^2,
-y0^2*y3-y0^2*y4+y0*y1*y2,
y0^2*y2-y0*y1*y2+y0*y1*y3+y0*y1*y4,
-y0^2*y4+y0*y1*y2-y0*y1*y3+y1^2*y3
],[
3*y0*y2^2*y4-3*y0*y3^2*y4+3*y0*y3*y4^2-3*y0*y4^3-3*y1*y2^2*y4-3*y1*y2*y3*y4-3*y1*y2*y4^2,
-3*y0^3*y4-3*y0^2*y1*y4-3*y2^3*y4-3*y2^2*y3*y4-3*y2^2*y4^2,
3*y0^2*y1*y4+3*y0*y1^2*y4+3*y2^3*y4-3*y2*y3^2*y4+3*y2*y3*y4^2-3*y2*y4^3,
y0^3*y2+y0^3*y3-2*y0^3*y4-3*y0^2*y1*y4+y1^3*y2+y1^3*y3+y1^3*y4+y2^4+y2^3*y3-2*y2^3*y4-
3*y2^2*y4^2+y2*y3^3+y2*y4^3+y3^4-2*y3^3*y4+3*y3^2*y4^2-2*y3*y4^3+y4^4
],[
3*y0^2*y2*y4-3*y0*y1*y2*y4+3*y1^2*y2*y4,
-3*y0*y2*y3*y4-3*y0*y2*y4^2+3*y1*y2^2*y4,
3*y0*y2^2*y4-3*y1*y2^2*y4+3*y1*y2*y3*y4+3*y1*y2*y4^2,
-y0^3*y1-3*y0*y2*y4^2-y1^4-y1*y2^3+3*y1*y2^2*y4-3*y1*y2*y3*y4-y1*y3^3-y1*y4^3
],[
-y0^2*y2*y3-y0^2*y2*y4-y0^2*y3^2+y0^2*y3*y4-y0^2*y4^2-y0*y1*y2^2+y0*y1*y3^2-y0*y1*y3*y4+
y0*y1*y4^2,
y0^2*y1^2+y0*y1^3+y0*y2*y3^2+2*y0*y2*y3*y4+y0*y2*y4^2+y0*y3^3+y0*y4^3+3*y1*y2*y3*y4,
y0^3*y1-y0*y1^3-y0*y2^2*y3-y0*y2^2*y4-y0*y2*y3^2+y0*y2*y3*y4-y0*y2*y4^2-3*y1*y2*y3*y4,
y0^2*y1^2+y0*y1^3+y0*y2*y3*y4+y0*y2*y4^2+y0*y3^2*y4-y0*y3*y4^2+y0*y4^3-y1*y2^2*y3+
3*y1*y2*y3*y4+y1*y3^3-y1*y3^2*y4+y1*y3*y4^2]];

//Below we give 4 lists [H,lambda,G], where lambda is a rational function in (a1,...,a4)
//and H,G are polynomials in X with coefficients that are rational functions in (a1,...,a4).
//The expression G^2+4*lambda*H^3 yields the same sextic in X for each triple [H,lambda,G].
//When (1:a1:a2:a3:a4) is a point on B that does not lie in a j-plane and has a4 != 0
//then this corresponds to the 3-torsion point corresponding to the j-plane y[0]=y[i]=0
//(for i=1,...,4, in the order given)
//(The article describes H,G as homogeneous forms in (x,z), of degrees 2 and 3 respectively.
//Here we set (x,z)=(X,1) to get a more compact representation)
HLGs:=[[(a1^3*a3*a4^3+a1^2*a2^2*a4^5+a1^2*a2^2*a4^2+2*a1*a2*a3^2*a4^4-a1*a2*a3^2*a4-a2^3*a3+
a3^4*a4^3)*X^2+(a1^4*a4^4+2*a1^2*a2*a3*a4^5-a1^2*a2*a3*a4^2+2*a1*a2^3*a4^4+a1*a2^3*a4+
2*a1*a3^3*a4^4+a1*a3^3*a4+2*a2^2*a3^2*a4^3+2*a2^2*a3^2)*X+a1^3*a2*a4^3+a1^2*a3^2*a4^5+
a1^2*a3^2*a4^2+2*a1*a2^2*a3*a4^4-a1*a2^2*a3*a4+a2^4*a4^3-a2*a3^3,
(-a4^3-1)/(a1^6*a4^6-6*a1^4*a2*a3*a4^4-2*a1^3*a2^3*a4^3-2*a1^3*a3^3*a4^3+9*a1^2*a2^2*a3^2*a4^2+
6*a1*a2^4*a3*a4+6*a1*a2*a3^4*a4+a2^6+2*a2^3*a3^3+a3^6),
(a1^6*a4^6+3*a1^4*a2*a3*a4^7-3*a1^4*a2*a3*a4^4+2*a1^3*a2^3*a4^9+4*a1^3*a2^3*a4^6+3*a1^3*a3^3*a4^6
+a1^3*a3^3*a4^3+6*a1^2*a2^2*a3^2*a4^8+3*a1^2*a2^2*a3^2*a4^5+6*a1^2*a2^2*a3^2*a4^2-
3*a1*a2^4*a3*a4^4+3*a1*a2^4*a3*a4+6*a1*a2*a3^4*a4^7+a2^6-3*a2^3*a3^3*a4^3-a2^3*a3^3+
2*a3^6*a4^6+a3^6*a4^3)/(a1^3*a4^3-3*a1*a2*a3*a4-a2^3-a3^3)*X^3+(3*a1^5*a2*a4^8+
3*a1^5*a2*a4^5+6*a1^4*a3^2*a4^7+6*a1^4*a3^2*a4^4+6*a1^3*a2^2*a3*a4^9+6*a1^3*a2^2*a3*a4^6+
6*a1^2*a2^4*a4^8+9*a1^2*a2^4*a4^5+3*a1^2*a2^4*a4^2+12*a1^2*a2*a3^3*a4^8+3*a1^2*a2*a3^3*a4^5-
9*a1^2*a2*a3^3*a4^2+12*a1*a2^3*a3^2*a4^7+9*a1*a2^3*a3^2*a4^4-3*a1*a2^3*a3^2*a4+6*a1*a3^5*a4^7+
6*a1*a3^5*a4^4-3*a2^5*a3*a4^3-3*a2^5*a3+6*a2^2*a3^4*a4^6+9*a2^2*a3^4*a4^3+
3*a2^2*a3^4)/(a1^3*a4^3-3*a1*a2*a3*a4-a2^3-a3^3)*X^2+(3*a1^5*a3*a4^8+3*a1^5*a3*a4^5+
6*a1^4*a2^2*a4^7+6*a1^4*a2^2*a4^4+6*a1^3*a2*a3^2*a4^9+6*a1^3*a2*a3^2*a4^6+12*a1^2*a2^3*a3*a4^8
+3*a1^2*a2^3*a3*a4^5-9*a1^2*a2^3*a3*a4^2+6*a1^2*a3^4*a4^8+9*a1^2*a3^4*a4^5+3*a1^2*a3^4*a4^2+
6*a1*a2^5*a4^7+6*a1*a2^5*a4^4+12*a1*a2^2*a3^3*a4^7+9*a1*a2^2*a3^3*a4^4-3*a1*a2^2*a3^3*a4+
6*a2^4*a3^2*a4^6+9*a2^4*a3^2*a4^3+3*a2^4*a3^2-3*a2*a3^5*a4^3-3*a2*a3^5)/(a1^3*a4^3-
3*a1*a2*a3*a4-a2^3-a3^3)*X+(a1^6*a4^6+3*a1^4*a2*a3*a4^7-3*a1^4*a2*a3*a4^4+3*a1^3*a2^3*a4^6
+a1^3*a2^3*a4^3+2*a1^3*a3^3*a4^9+4*a1^3*a3^3*a4^6+6*a1^2*a2^2*a3^2*a4^8+3*a1^2*a2^2*a3^2*a4^5
+6*a1^2*a2^2*a3^2*a4^2+6*a1*a2^4*a3*a4^7-3*a1*a2*a3^4*a4^4+3*a1*a2*a3^4*a4+2*a2^6*a4^6+
a2^6*a4^3-3*a2^3*a3^3*a4^3-a2^3*a3^3+a3^6)/(a1^3*a4^3-3*a1*a2*a3*a4-a2^3-a3^3)
],[
a1*a4*X^2+a2*X-a3,
-a1^3*a4^9-a1^3*a4^6+3*a1*a2*a3*a4^7+3*a1*a2*a3*a4^4+a2^3*a4^6+a2^3*a4^3+a3^3*a4^6+a3^3*a4^3,
(2*a1^3*a4^6+a1^3*a4^3-3*a1*a2*a3*a4^4+a2^3-a3^3*a4^3)*X^3+(3*a1^2*a2*a4^5+3*a1^2*a2*a4^2-
3*a2^2*a3*a4^3-3*a2^2*a3)*X^2+(-3*a1^2*a3*a4^5-3*a1^2*a3*a4^2+3*a2*a3^2*a4^3+3*a2*a3^2)*X-
a1^3*a4^3-3*a1*a2*a3*a4^4-a2^3*a4^3-2*a3^3*a4^3-a3^3
],[
a2*X^2-a3*X-a1*a4,
a1^3*a4^9+a1^3*a4^6-3*a1*a2*a3*a4^7-3*a1*a2*a3*a4^4-a2^3*a4^6-a2^3*a4^3-a3^3*a4^6-a3^3*a4^3,
(a1^3*a4^3+3*a1*a2*a3*a4^4+2*a2^3*a4^3+a2^3+a3^3*a4^3)*X^3+(3*a1^2*a2*a4^5+3*a1^2*a2*a4^2-
3*a2^2*a3*a4^3-3*a2^2*a3)*X^2+(-3*a1^2*a3*a4^5-3*a1^2*a3*a4^2+3*a2*a3^2*a4^3+3*a2*a3^2)*X-
2*a1^3*a4^6-a1^3*a4^3+3*a1*a2*a3*a4^4+a2^3*a4^3-a3^3
],[
(a1*a2^2*a4^3+a1*a2^2)*X^2+(a1^3*a4^2+a1*a2*a3*a4^3-2*a1*a2*a3+a2^3*a4^2+a3^3*a4^2)*X+
a1*a3^2*a4^3+a1*a3^2,
(-a1^3*a4^3+3*a1*a2*a3*a4+a2^3+a3^3)/(a1^6+6*a1^4*a2*a3*a4+2*a1^3*a2^3+2*a1^3*a3^3+
9*a1^2*a2^2*a3^2*a4^2+6*a1*a2^4*a3*a4+6*a1*a2*a3^4*a4+a2^6+2*a2^3*a3^3+a3^6),
(a1^6*a4^3+6*a1^4*a2*a3*a4^4+2*a1^3*a2^3*a4^6+5*a1^3*a2^3*a4^3+a1^3*a2^3+2*a1^3*a3^3*a4^3+
9*a1^2*a2^2*a3^2*a4^5+3*a1*a2^4*a3*a4^4-3*a1*a2^4*a3*a4+6*a1*a2*a3^4*a4^4-a2^6+a2^3*a3^3*a4^3
-a2^3*a3^3+a3^6*a4^3)/(a1^3+3*a1*a2*a3*a4+a2^3+a3^3)*X^3+(3*a1^5*a2*a4^5+3*a1^5*a2*a4^2+
3*a1^3*a2^2*a3*a4^6-3*a1^3*a2^2*a3+3*a1^2*a2^4*a4^5+3*a1^2*a2^4*a4^2+3*a1^2*a2*a3^3*a4^5+
3*a1^2*a2*a3^3*a4^2+9*a1*a2^3*a3^2*a4^4+9*a1*a2^3*a3^2*a4+3*a2^5*a3*a4^3+3*a2^5*a3+
3*a2^2*a3^4*a4^3+3*a2^2*a3^4)/(a1^3+3*a1*a2*a3*a4+a2^3+a3^3)*X^2+(-3*a1^5*a3*a4^5-
3*a1^5*a3*a4^2-3*a1^3*a2*a3^2*a4^6+3*a1^3*a2*a3^2-3*a1^2*a2^3*a3*a4^5-3*a1^2*a2^3*a3*a4^2-
3*a1^2*a3^4*a4^5-3*a1^2*a3^4*a4^2-9*a1*a2^2*a3^3*a4^4-9*a1*a2^2*a3^3*a4-3*a2^4*a3^2*a4^3-
3*a2^4*a3^2-3*a2*a3^5*a4^3-3*a2*a3^5)/(a1^3+3*a1*a2*a3*a4+a2^3+a3^3)*X+(-a1^6*a4^3-
6*a1^4*a2*a3*a4^4-2*a1^3*a2^3*a4^3-2*a1^3*a3^3*a4^6-5*a1^3*a3^3*a4^3-a1^3*a3^3-
9*a1^2*a2^2*a3^2*a4^5-6*a1*a2^4*a3*a4^4-3*a1*a2*a3^4*a4^4+3*a1*a2*a3^4*a4-a2^6*a4^3-
a2^3*a3^3*a4^3+a2^3*a3^3+a3^6)/(a1^3+3*a1*a2*a3*a4+a2^3+a3^3)]];

//Below is a triple (H,lambda,G) such that G^2+4*lambda*H^3 is -3*F, where F is the sextic
//defined by HLGs above. This triple corresponds to the 3-torsion points that the j-plane
//y0+...+y4=y0+y4=0 marks if (1:a1:a2:a3:a4) is a point on the Burkhardt quartic. Note that
//the identity of the sextics holds regardless of whether (1:a1:a2:a3:a4) satisfy the Burkhardt
//relation. We do need the Burkhardt relation to get the other cyclic order 3 subgroups defined
//over the base field.
HLGdual:=[(a1^2*a4^2+a1*a2*a4^3-a1*a2*a4^2-a1*a2*a4-a1*a3*a4^2+a2^2+a2*a3*a4+a3^2*a4^2)*X^2+(-a1^2*a4^3+
a1^2*a4^2+a1*a2*a4^3+a1*a2*a4+a1*a3*a4^3+a1*a3*a4+a2^2*a4^2-a2^2*a4-2*a2*a3+a3^2*a4^2-
a3^2*a4)*X+a1^2*a4^2-a1*a2*a4^2+a1*a3*a4^3-a1*a3*a4^2-a1*a3*a4+a2^2*a4^2+a2*a3*a4+a3^2,
(-3*a1^2*a4^4+3*a1^2*a4^3-3*a1^2*a4^2-3*a1*a2*a4^3+3*a1*a2*a4^2-3*a1*a2*a4-3*a1*a3*a4^3+
3*a1*a3*a4^2-3*a1*a3*a4-3*a2^2*a4^2+3*a2^2*a4-3*a2^2+3*a2*a3*a4^2-3*a2*a3*a4+3*a2*a3-
3*a3^2*a4^2+3*a3^2*a4-3*a3^2)/(a1^2*a4^4+2*a1^2*a4^3+a1^2*a4^2-2*a1*a2*a4^3-4*a1*a2*a4^2-
2*a1*a2*a4-2*a1*a3*a4^3-4*a1*a3*a4^2-2*a1*a3*a4+a2^2*a4^2+2*a2^2*a4+a2^2+2*a2*a3*a4^2+
4*a2*a3*a4+2*a2*a3+a3^2*a4^2+2*a3^2*a4+a3^2),
(-3*a1^4*a4^5+3*a1^4*a4^4-6*a1^3*a2*a4^6+6*a1^3*a2*a4^5-3*a1^3*a2*a4^4-3*a1^3*a2*a4^3+
6*a1^3*a3*a4^5-3*a1^3*a3*a4^4+3*a1^3*a3*a4^3+6*a1^2*a2^2*a4^6+6*a1^2*a2^2*a4^4+6*a1^2*a2^2*a4^2+
3*a1^2*a2*a3*a4^6-3*a1^2*a2*a3*a4^5+6*a1^2*a2*a3*a4^4+6*a1^2*a2*a3*a4^3-6*a1^2*a2*a3*a4^2-
6*a1^2*a3^2*a4^5+6*a1^2*a3^2*a4^4-6*a1^2*a3^2*a4^3-6*a1*a2^3*a4^4+6*a1*a2^3*a4^3-3*a1*a2^3*a4^2-
3*a1*a2^3*a4-3*a1*a2^2*a3*a4^5+3*a1*a2^2*a3*a4^4-6*a1*a2^2*a3*a4^3-6*a1*a2^2*a3*a4^2+
6*a1*a2^2*a3*a4+9*a1*a2*a3^2*a4^5-3*a1*a2*a3^2*a4^4-6*a1*a2*a3^2*a4^3+6*a1*a2*a3^2*a4^2+
3*a1*a3^3*a4^5-3*a1*a3^3*a4^4+6*a1*a3^3*a4^3-3*a2^4*a4+3*a2^4-6*a2^3*a3*a4^2+3*a2^3*a3*a4-
3*a2^3*a3-6*a2^2*a3^2*a4^3+6*a2^2*a3^2*a4^2-6*a2^2*a3^2*a4-3*a2*a3^3*a4^4+3*a2*a3^3*a4^3-
6*a2*a3^3*a4^2+3*a3^4*a4^4-3*a3^4*a4^3)/(a1*a4^2+a1*a4-a2*a4-a2-a3*a4-a3)*X^3+(6*a1^4*a4^6
-6*a1^4*a4^5+6*a1^4*a4^4+3*a1^3*a2*a4^7-9*a1^3*a2*a4^6+12*a1^3*a2*a4^5-9*a1^3*a2*a4^4+
3*a1^3*a2*a4^3-6*a1^3*a3*a4^6+12*a1^3*a3*a4^5-12*a1^3*a3*a4^4+6*a1^3*a3*a4^3+3*a1^2*a2^2*a4^6-
15*a1^2*a2^2*a4^5+18*a1^2*a2^2*a4^4-15*a1^2*a2^2*a4^3+3*a1^2*a2^2*a4^2+9*a1^2*a2*a3*a4^6-
9*a1^2*a2*a3*a4^5+9*a1^2*a2*a3*a4^3-9*a1^2*a2*a3*a4^2+6*a1^2*a3^2*a4^6-12*a1^2*a3^2*a4^5+
18*a1^2*a3^2*a4^4-12*a1^2*a3^2*a4^3+6*a1^2*a3^2*a4^2+12*a1*a2^3*a4^5-6*a1*a2^3*a4^4+
12*a1*a2^3*a4^3+6*a1*a2^3*a4+3*a1*a2^2*a3*a4^5+3*a1*a2^2*a3*a4^4+3*a1*a2^2*a3*a4^2+
3*a1*a2^2*a3*a4+6*a1*a2*a3^2*a4^5-12*a1*a2*a3^2*a4^4+6*a1*a2*a3^2*a4^2-12*a1*a2*a3^2*a4+
6*a1*a3^3*a4^5-12*a1*a3^3*a4^4+12*a1*a3^3*a4^3-6*a1*a3^3*a4^2-6*a2^4*a4^3+6*a2^4*a4^2-6*a2^4*a4
-3*a2^3*a3*a4^4+3*a2^3*a3*a4^3-12*a2^3*a3*a4^2+9*a2^3*a3*a4-9*a2^3*a3+9*a2^2*a3^2*a4^4-
9*a2^2*a3^2*a4^3+18*a2^2*a3^2*a4^2-9*a2^2*a3^2*a4+9*a2^2*a3^2+12*a2*a3^3*a4^3-12*a2*a3^3*a4^2+
12*a2*a3^3*a4+6*a3^4*a4^4-6*a3^4*a4^3+6*a3^4*a4^2)/(a1*a4^2+a1*a4-a2*a4-a2-a3*a4-a3)*X^2+
(6*a1^4*a4^6-6*a1^4*a4^5+6*a1^4*a4^4-6*a1^3*a2*a4^6+12*a1^3*a2*a4^5-12*a1^3*a2*a4^4+
6*a1^3*a2*a4^3+3*a1^3*a3*a4^7-9*a1^3*a3*a4^6+12*a1^3*a3*a4^5-9*a1^3*a3*a4^4+3*a1^3*a3*a4^3+
6*a1^2*a2^2*a4^6-12*a1^2*a2^2*a4^5+18*a1^2*a2^2*a4^4-12*a1^2*a2^2*a4^3+6*a1^2*a2^2*a4^2+
9*a1^2*a2*a3*a4^6-9*a1^2*a2*a3*a4^5+9*a1^2*a2*a3*a4^3-9*a1^2*a2*a3*a4^2+3*a1^2*a3^2*a4^6-
15*a1^2*a3^2*a4^5+18*a1^2*a3^2*a4^4-15*a1^2*a3^2*a4^3+3*a1^2*a3^2*a4^2+6*a1*a2^3*a4^5-
12*a1*a2^3*a4^4+12*a1*a2^3*a4^3-6*a1*a2^3*a4^2+6*a1*a2^2*a3*a4^5-12*a1*a2^2*a3*a4^4+
6*a1*a2^2*a3*a4^2-12*a1*a2^2*a3*a4+3*a1*a2*a3^2*a4^5+3*a1*a2*a3^2*a4^4+3*a1*a2*a3^2*a4^2+
3*a1*a2*a3^2*a4+12*a1*a3^3*a4^5-6*a1*a3^3*a4^4+12*a1*a3^3*a4^3+6*a1*a3^3*a4+6*a2^4*a4^4-
6*a2^4*a4^3+6*a2^4*a4^2+12*a2^3*a3*a4^3-12*a2^3*a3*a4^2+12*a2^3*a3*a4+9*a2^2*a3^2*a4^4-
9*a2^2*a3^2*a4^3+18*a2^2*a3^2*a4^2-9*a2^2*a3^2*a4+9*a2^2*a3^2-3*a2*a3^3*a4^4+3*a2*a3^3*a4^3-
12*a2*a3^3*a4^2+9*a2*a3^3*a4-9*a2*a3^3-6*a3^4*a4^3+6*a3^4*a4^2-6*a3^4*a4)/(a1*a4^2+a1*a4-
a2*a4-a2-a3*a4-a3)*X+(-3*a1^4*a4^5+3*a1^4*a4^4+6*a1^3*a2*a4^5-3*a1^3*a2*a4^4+3*a1^3*a2*a4^3
-6*a1^3*a3*a4^6+6*a1^3*a3*a4^5-3*a1^3*a3*a4^4-3*a1^3*a3*a4^3-6*a1^2*a2^2*a4^5+6*a1^2*a2^2*a4^4-
6*a1^2*a2^2*a4^3+3*a1^2*a2*a3*a4^6-3*a1^2*a2*a3*a4^5+6*a1^2*a2*a3*a4^4+6*a1^2*a2*a3*a4^3-
6*a1^2*a2*a3*a4^2+6*a1^2*a3^2*a4^6+6*a1^2*a3^2*a4^4+6*a1^2*a3^2*a4^2+3*a1*a2^3*a4^5-
3*a1*a2^3*a4^4+6*a1*a2^3*a4^3+9*a1*a2^2*a3*a4^5-3*a1*a2^2*a3*a4^4-6*a1*a2^2*a3*a4^3+
6*a1*a2^2*a3*a4^2-3*a1*a2*a3^2*a4^5+3*a1*a2*a3^2*a4^4-6*a1*a2*a3^2*a4^3-6*a1*a2*a3^2*a4^2+
6*a1*a2*a3^2*a4-6*a1*a3^3*a4^4+6*a1*a3^3*a4^3-3*a1*a3^3*a4^2-3*a1*a3^3*a4+3*a2^4*a4^4-
3*a2^4*a4^3-3*a2^3*a3*a4^4+3*a2^3*a3*a4^3-6*a2^3*a3*a4^2-6*a2^2*a3^2*a4^3+6*a2^2*a3^2*a4^2-
6*a2^2*a3^2*a4-6*a2*a3^3*a4^2+3*a2*a3^3*a4-3*a2*a3^3-3*a3^4*a4+3*a3^4)/(a1*a4^2+a1*a4-a2*a4-
a2-a3*a4-a3)];

//magma code to verify that the expressions given indeed satisfy the relations claimed,
L:=[[Evaluate(p,phi):p in q]:q in psis];
assert {[Pt|c/l[1]: c in l]:l in L} eq{[1,t1,t2,t3]};
assert Evaluate(B,phi) eq 0;
V:={c[3]^2+4*c[2]*c[1]^3:c in HLGs};
assert #V eq 1;
F:=Rep(V);
G:=c[3]^2+4*c[2]*c[1]^3 where c:=HLGdual;
assert G/F eq -3;