Let C be the curve with projective equation (Z^2) 
          (Y^2) = U(X, Y)  =  (Subscript[a, 0]) (X^4) + 4 (Subscript[a, 1]) (X^3) Y + 
            6  (Subscript[a, 2]) (X^2) (Y^2) + 
            4  (Subscript[a, 3]) X (Y^3) + (Subscript[a, 4]) (Y^4). 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 equation on the line below.

a0 = 0; a1 = 1; a2 = 0; a3 = 1; a4 = 1

Now we set up the invariants and covariants as defined in the article:

U = 
        a0 X^4  +  4 a1 X^3 Y  +  6 a2 X^2 Y^2  +  
          4 a3 X Y^3  + a4 Y^4;

g =   Expand[(1/(2^4 3^2)) (D[U, X, Y]^2 - 
                D[U, {X, 2}] D[U, {Y, 2}])];

h = 
        Expand[(1/2^3) (D[U, X] D[g, Y] - 
                D[g, X] D[U, Y])];

i =  a0 a4  -  4 a1 a3  +  3  a2^2;

j =  
        a0 a2 a4  +  2 a1 a2 a3  -  a0 a3^2  -  a4 a1^2  -  
          a2^3;

Uncomment the following if you want to test that the syzygy is satisfied.

Now we convert to the notation set up at the beginning of this file:

g2 = i; g3 = j; x =  g Z Y; y =  h;  z  =  Z^3 Y^3;

Here are the coefficients of the jacobian and the equations for the covering map:

g2

g3

x

y

z

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, 
      Y^2 Z^2  -  U]