Selected formulae for the genus one sextic curve \begin{equation} \label{eq:curve} y^3 \enspace = \enspace \left( ax^3 \enspace + \enspace bx^2 \enspace + \enspace cx \enspace + \enspace d \right)^2 \end{equation}
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 \begin{equation} \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 \end{equation} where $\discrim$ denotes the discriminant of the third degree polynomials.
Let $f$ be a solution of the differential equation \begin{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 \end{equation} with the boundary condition $f'(0) = 0$. Then $f$ is a three-fold symmetric function and it's cross-ratio satisfies the formula \begin{equation} \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)} \end{equation} 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 \begin{equation} 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)} \end{equation} Then $f$ has an addition formula given by \begin{equation} 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}} \end{equation} and $g$ has an addition formula given by \begin{equation} 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}} \end{equation} 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 \eqref{eq:curve}.
The "Abel Sum" \begin{equation} \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 \end{equation} has an algebraic integral given by \begin{equation} \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 \end{equation} where $x_3$ is the constant of integration.