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Selected formulae for the genus one sextic curve
y^3 \enspace = \enspace \left( ax^3 \enspace + \enspace bx^2 \enspace + \enspace cx \enspace + \enspace d \right)^2
If the roots of two cubic polynomials are mapped to one another by a Möbius transform $L$ then the change of variables $x = L(u)$ gives
\sqrt[6] {\discrim(a,b,c,d)} \space \bigint \frac 1 {\left(ax^3 + bx^2 + cx + d\right)^{\sfrac 2 3}} dx \enspace = \enspace
\sqrt[6] {\discrim(p,q,r,s)} \space \bigint \frac 1 {\left(pu^3 + qu^2 + ru + s\right)^{\sfrac 2 3}} du
where $\discrim$ denotes the discriminant of the third degree polynomials.
Let $f$ be a solution of the differential equation
f'(z)^3 \enspace = \enspace \left[ a f(z)^3 \space + \space b f(z)^2 \space + \space c f(z) \space + \space d \right]^2
with the boundary condition $f'(0) = 0$.
Then $f$ is a three-fold symmetric function and it's cross-ratio satisfies the formula
\frac {\big[f(z_1)-f(z_2)\big]\thinspace\big[f(z_3)-f(z_4)\big]} {\big[f(z_1)-f(z_3)\big]\thinspace\big[f(z_2)-f(z_4)\big]} \enspace = \enspace \frac
{\sigma(z_1-z_2) \thinspace \sigma(z_1-\zeta z_2) \thinspace \sigma(z_1-\zeta^2 z_2) \thinspace \sigma(z_3-z_4) \thinspace \sigma(z_3-\zeta z_4) \thinspace \sigma(z_3-\zeta^2 z_4)}
{\sigma(z_1-z_3) \thinspace \sigma(z_1-\zeta z_3) \thinspace \sigma(z_1-\zeta^2 z_3) \thinspace \sigma(z_2-z_4) \thinspace \sigma(z_2-\zeta z_4) \thinspace \sigma(z_2-\zeta^2 z_4)}
where $\zeta$ is a primitive third root of unity and $\sigma$ is the Weierstrass sigma
function with the same periods as $f$.
The function $f'(z)$ is a perfect square, let
g(z) \enspace = \enspace \sqrt{f'(z)} \enspace = \enspace \frac {a f(z)^3 \space + \space b f(z)^2 \space + \space c f(z) \space + \space d} {f'(z)}
Then $f$ has an addition formula given by
f(u + v) \enspace = \enspace \frac 1 {f(u) f(v)} \frac {
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
1 & 0 & \sqrt[3]{d} \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
1 & 0 & \zeta\sqrt[3]{d} \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
1 & 0 & \zeta^2\sqrt[3]{d} \\
\end{vmatrix}}
{\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
0 & 1 & \sqrt[3]{a} \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
0 & 1 & \zeta\sqrt[3]{a} \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
0 & 1 & \zeta^2\sqrt[3]{a} \\
\end{vmatrix}}
and $g$ has an addition formula given by
g(u + v) \enspace = \enspace \frac {a} {g(u) g(v)} \frac {
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
1 & e_1 & 0 \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
1 & e_2 & 0 \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
1 & e_3 & 0 \\
\end{vmatrix}}
{\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
0 & 1 & \sqrt[3]{a} \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
0 & 1 & \zeta\sqrt[3]{a} \\
\end{vmatrix} \space
\begin{vmatrix}
1 & f(u) & g(u) \\
1 & f(v) & g(v) \\
0 & 1 & \zeta^2\sqrt[3]{a} \\
\end{vmatrix}}
where $\zeta$ is a primitive third root of unity and $e_1,e_2,e_3$ are the three roots of the right hand side of curve.
The "Abel Sum"
\frac {dx_1} {\left[(x_1-e_1)(x_1-e_2)(x_1-e_3)\right]^{\sfrac 2 3}} \enspace + \enspace
\frac {dx_2} {\left[(x_2-e_1)(x_2-e_2)(x_2-e_3)\right]^{\sfrac 2 3}} \enspace = \enspace 0
has an algebraic integral given by
\begin{vmatrix}
\sqrt[3]{x_1 - e_1} & \sqrt[3]{x_1 - e_2} & \sqrt[3]{x_1 - e_3} \\
\sqrt[3]{x_2 - e_1} & \sqrt[3]{x_2 - e_2} & \sqrt[3]{x_2 - e_3} \\
\sqrt[3]{x_3 - e_1} & \sqrt[3]{x_3 - e_2} & \sqrt[3]{x_3 - e_3} \\
\end{vmatrix}
\enspace = \enspace 0
where $x_3$ is the constant of integration.