Let C be the plane cubic defined by:

 U(X, Y, 
          Z) = a (X^3) + b (Y^3) + c (Z^3) + 3 (Subscript[a, 2]) (X^2) Y + 
            3  (Subscript[a, 3]) (X^2) Z + 3 (Subscript[b, 1]) (Y^2) X + 
            3 (Subscript[b, 3]) (Y^2) Z + 3 (Subscript[c, 1]) (Z^2) X + 
            3 (Subscript[c, 2]) (Z^2) Y + 6 m X Y Z  =  0.
 
Below we give the jacobian E and the covering map (Subscript[ϕ, D]) :  C → E in terms of the equation (y^2) z =  
        4 (x^3) - (Subscript[g, 2]) x (z^2) - (Subscript[g, 3]) (z^3) for E.
Enter the coefficients of your form on the line below

a  =  13; b = 19; c = -11;
a2 = 32; a3 = 7; b1 = -90; b3 = 33; c1 = -5; c2 = 4; 
    m = 56;

Now we compute the invariants.

S = 
        a b c m  -   (b c a2 a3  +  c a b1 b3  +  
              a b c1 c2 )  -  
          m (a b3 c2  +  b c1 a3  +  
                c a2 b1 )  +  (a b1 (c2^2) +  a c1 (b3^2) + 
              b a2 (c1^2) +  b c2 (a3^2)  +  c b3 (a2^2) +  
              c a3 (b1^2)) - (m^4)  +  
          2 (m^2) (b1 c1  +  c2 a2  + a3 b3 )  -  
          3 m (a2 b3 c1  + a3 b1 c2 )  -  ((b1^2) 
                (c1^2) + (c2^2) (a2^2) + (a3^2) 
                (b3^2))  +  (c2 a2 a3 b3  + a3 b3 b1 c1  + 
              b1 c1 c2 a2 );

\(\(T = 
        8*a3^3*b3^3\  + \ 4*a3^3*b^2*c\  - \ 12*a2*a3^2*b*b3*c\  + \ 
          24*a3^2*b1^2*b3*c\  + \ 
\ \ 24*a2^2*a3*b3^2*c\  - \ 
          12*a*a3*b1*b3^2*c\  + \ 4*a^2*b3^3*c\  + \ 
          4*a2^3*b*c^2\  + \ 
\ \ a^2*b^2*c^2\  - \ 6*a*a2*b*b1*c^2\  - \ 
          3*a2^2*b1^2*c^2\  + \ 
          4*a*b1^3*c^2\  - \ 
\ \ 12*a3^2*b1*b3^2*c1\  - \ 
          12*a*a3*b3^3*c1\  - \ 6*a*a3*b^2*c*c1\  + \ 
          18*a2*a3*b*b1*c*c1\  - \ 
\ \ 12*a3*b1^3*c*c1\  + \ 
          6*a*a2*b*b3*c*c1\  + \ 6*a2^2*b1*b3*c*c1\  - \ 
          12*a*b1^2*b3*c*c1\  - \ 
\ \ 3*a3^2*b^2*c1^2\  + \ 
          6*a2*a3*b*b3*c1^2\  - \ 
          12*a3*b1^2*b3*c1^2\  - \ 
\ \ 27*a2^2*b3^2*c1^2\  + \ 
          24*a*b1*b3^2*c1^2\  + \ 4*a*b^2*c1^3\  - \ 
          12*a2*b*b1*c1^3\  + \ 
\ \ 8*b1^3*c1^3\  - \ 12*a3^3*b*b3*c2\  - \ 
          12*a2*a3^2*b3^2*c2\  - \ 
          12*a2^2*a3*b*c*c2\  + \ 
\ \ 6*a*a3*b*b1*c*c2\  + \ 
          6*a2*a3*b1^2*c*c2\  - \ 12*a2^3*b3*c*c2\  - \ 
          6*a^2*b*b3*c*c2\  + \ 
\ \ 18*a*a2*b1*b3*c*c2\  + \ 
          6*a3^2*b*b1*c1*c2\  + \ 
          18*a*a3*b*b3*c1*c2\  - \ 
\ \ 6*a2*a3*b1*b3*c1*c2\  + \ 
          6*a*a2*b3^2*c1*c2\  + \ 
          24*a2^2*b*c1^2*c2\  - \ 
\ \ 12*a*b*b1*c1^2*c2\  - \ 
          12*a2*b1^2*c1^2*c2\  + \ 
          24*a2*a3^2*b*c2^2\  - \ 
\ \ 27*a3^2*b1^2*c2^2\  - \ 
          12*a2^2*a3*b3*c2^2\  + \ 
          6*a*a3*b1*b3*c2^2\  - \ 
\ \ 3*a^2*b3^2*c2^2\  - \ 
          12*a*a2*b*c1*c2^2\  - \ 
          12*a2^2*b1*c1*c2^2\  + \ 
\ \ 24*a*b1^2*c1*c2^2\  + \ 
          8*a2^3*c2^3\  + \ 4*a^2*b*c2^3\  - \ 
          12*a*a2*b1*c2^3\  - \ 
\ \ 24*a3^2*b*b1*c*m\  + \ 
          12*a*a3*b*b3*c*m\  - \ 60*a2*a3*b1*b3*c*m\  - \ 
          24*a*a2*b3^2*c*m\  + \ 
\ \ 12*a3^2*b*b3*c1*m\  + \ 
          36*a2*a3*b3^2*c1*m\  - \ 
          24*a2^2*b*c*c1*m\  + \ 
\ \ 12*a*b*b1*c*c1*m\  + \ 
          12*a2*b1^2*c*c1*m\  + \ 12*a3*b*b1*c1^2*m\  - \ 
          24*a*b*b3*c1^2*m\  + \ 
\ \ 36*a2*b1*b3*c1^2*m\  + \ 
          36*a3^2*b1*b3*c2*m\  + \ 
          12*a*a3*b3^2*c2*m\  + \ 
\ \ 12*a*a2*b*c*c2*m\  + \ 
          12*a2^2*b1*c*c2*m\  - \ 
          24*a*b1^2*c*c2*m\  - \ 
\ \ 60*a2*a3*b*c1*c2*m\  + \ 
          36*a3*b1^2*c1*c2*m\  + \ 
          36*a2^2*b3*c1*c2*m\  - \ 
\ \ 60*a*b1*b3*c1*c2*m\  - \ 
          24*a*a3*b*c2^2*m\  + \ 
          36*a2*a3*b1*c2^2*m\  + \ 
\ \ 12*a*a2*b3*c2^2*m\  - \ 
          24*a3^2*b3^2*m^2\  + \ 36*a2*a3*b*c*m^2\  + \ 
          12*a3*b1^2*c*m^2\  + \ 
\ \ 12*a2^2*b3*c*m^2\  + \ 
          36*a*b1*b3*c*m^2\  - \ 12*a3*b1*b3*c1*m^2\  + \ 
          12*a*b3^2*c1*m^2\  + \ 
\ \ 12*a2*b*c1^2*m^2\  - \ 
          24*b1^2*c1^2*m^2\  + \ 12*a3^2*b*c2*m^2\  - \ 
          12*a2*a3*b3*c2*m^2\  + \ 
\ \ 36*a*b*c1*c2*m^2\  - \ 
          12*a2*b1*c1*c2*m^2\  - \ 24*a2^2*c2^2*m^2\  + \ 
          12*a*b1*c2^2*m^2\  - \ 
\ \ 20*a*b*c*m^3\  - \ 
          12*a2*b1*c*m^3\  - \ 12*a3*b*c1*m^3\  - \ 
          36*a2*b3*c1*m^3\  - \ 
\ \ 36*a3*b1*c2*m^3\  - \ 
          12*a*b3*c2*m^3\  + \ 24*a3*b3*m^4\  + \ 
          24*b1*c1*m^4\  + \ 
\ \ 24*a2*c2*m^4\  - \ 8*m^6;\)\)

g2 = -108 S; g3 = 27 T;

Here are the coefficients of the jacobian:

g2

g3

Now we compute the covariants as in the paper. Ignore the error message about spelling errors.

First the easy ones, U and H.

U = 
        a X^3  +  b Y^3  +  c Z^3  +  3 a2  X^2 Y  +  
          3 a3  X^2  Z  +  3 b1  Y^2 X  +  3 b3  Y^2 Z  +  
          3 c1  Z^2 X  + 3 c2 Z^2 Y  +  6 m X Y Z;

\(\(H\  = 
        Expand[Det[{{D[U, {X, 2}], D[U, X, Y], D[U, X, Z]}, 
	{D[U, Y, X], 
                  D[U, {Y, 2}], D[U, Y, Z]}, 
		{D[U, Z, X], D[U, Z, Y], 
                  D[U, {Z, 2}]}}]/216];\)\)

Now we compute the polar conic of U and save its coefficients

PU = Expand[X' D[U, X] + Y' D[U, Y] + Z' D[U, Z]];

PUa = Coefficient[PU, X^2]; PUb = Coefficient[PU, Y^2]; 
    PUc = Coefficient[PU, Z^2]; 
    PUf = Simplify[(1/2) Coefficient[PU, Y Z]]; 
    PUg = Simplify[(1/2) Coefficient[PU, Z X]]; 
    PUh = Simplify[(1/2) Coefficient[PU, X Y]];

And next, the polar conic of H

PH = Expand[X' D[H, X] + Y' D[H, Y] + Z' D[H, Z]];

PHa = Coefficient[PH, X^2]; PHb = Coefficient[PH, Y^2]; 
    PHc = Coefficient[PH, Z^2]; 
    PHf = Simplify[(1/2) Coefficient[PH, Y Z]]; 
    PHg = Simplify[(1/2) Coefficient[PH, Z X]]; 
    PHh = Simplify[(1/2) Coefficient[PH, X Y]];

Next we find the coefficients of the duals of these polar conics.
First PU

PUA  =  PUb PUc  -  PUf^2; PUB =  PUc PUa  -  PUg^2; 
    PUC  =  PUa PUb  -  PUh^2; PUF = PUg PUh  -  PUa PUf ; 
    PUG  =  PUh PUf  -  PUb PUg; 
    PUH  =  PUf PUg  -  PUc PUh;

Then PH

PHA  =  PHb PHc  -  PHf^2; PHB =  PHc PHa  -  PHg^2; 
    PHC  =  PHa PHb  -  PHh^2; PHF = PHg PHh  -  PHa PHf ; 
    PHG  =  PHh PHf  -  PHb PHg; 
    PHH =  PHf PHg  -  PHc PHh;

Finally, we calculate Θ and J

Θ = 
        Expand[(1/
                  9) ((PUB PHC  +  PUC PHB  -  
                        2 PUF PHF) X^2  +  (PUC PHA  +  
                        PHC PUA  -  
                        2 PUG PHG) Y^2  +  (PUA PHB  +  
                        PHA PUB  -  2 PUH PHH) Z^2  +  
                  2 (PUG PHH  +  PHG PUH  -  PUA PHF  -  
                        PHA PUF) Y Z  +  
                  2 ( PUH PHF  +  PHH PUF  -  PUB PHG  -  
                        PHB PUG) Z X  +  
                  2 (PUF PHG  +  PHF PUG  -  PUC PHH  -  
                        PHC PUH) X Y) /. {X' ->  X,  Y'  ->  Y,  
              Z' ->  Z}];

J = 
        Expand[-(1/9) 
            Det[{{D[U, X], D[U, Y], D[U, Z]}, {D[H, X], D[H, Y], 
                  D[H, Z]}, {D[Θ, X], D[Θ, Y], 
                  D[Θ, Z]}}]];

Uncomment the following if you want to check the syzygy.

Here are the equations for the covering map.

x = Expand[Θ H]; y = J; z = Expand[H^3];

The following calculation checks that the given equations define a map from C to E.

PolynomialMod[y^2 z  -  4 x^3  +  g2 x z^2  +  g3 z^3, 
      U]