Selected formulae for the genus one quartic curve \begin{equation} \label{eq:curve} y^2 \enspace = \enspace ax^4 \enspace + \enspace bx^3 \enspace + \enspace cx^2 \enspace + \enspace dx \enspace + \enspace e \end{equation}
If the roots of two quartic polynomials can be mapped to one another by a Möbius transform $L$ then the change of variables $x = L(u)$ gives \begin{equation} \sqrt[12] {\discrim(a,b,c,d,e)} \space \bigint \frac 1 {\sqrt{ax^4 + bx^3 + cx^2 + dx + e}} dx \enspace = \enspace \sqrt[12] {\discrim(p,q,r,s,t)} \space \bigint \frac 1 {\sqrt{pu^4 + qu^3 + ru^2 + su + t}} du \end{equation} where $\discrim$ denotes the discriminant of the fourth degree polynomials.
Let $f$ be a solution of the differential equation \begin{equation} f'(z)^2 \enspace = \enspace a f(z)^4 \space + \space b f(z)^3 \space + \space c f(z)^2 \space + \space d f(z) \space + \space e \end{equation} with the boundary condition $f'(0) = 0$. Then $f$ is an even 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)\sigma(z_1+z_2)\sigma(z_3-z_4)\sigma(z_3+z_4)} {\sigma(z_1-z_3)\sigma(z_1+z_3)\sigma(z_2-z_4)\sigma(z_2+z_4)} \end{equation} where $\sigma$ is the Weierstrass sigma function with the same periods as $f$.
Let $g$ be a any solution of the differential equation \begin{equation} g'(z)^2 \enspace = \enspace a g(z)^4 \space + \space b g(z)^3 \space + \space c g(z)^2 \space + \space d g(z) \space + \space e \end{equation} Then $g$ has an addition formula given by \begin{equation} g(u + v) \enspace = \enspace \frac 1 {g(u) g(v) g(0)} \frac {\begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 1 & 0 & 0 & \sqrt{e} \\ \end{vmatrix} \space \begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 1 & 0 & 0 & -\sqrt{e} \\ \end{vmatrix}} {\begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 0 & 0 & 1 & \sqrt{a} \\ \end{vmatrix} \space \begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 0 & 0 & 1 & -\sqrt{a} \\ \end{vmatrix}} \end{equation} and $g'$ has an addition formula given by \begin{equation} g'(u + v) \enspace = \enspace -\frac {a^2} {g'(u) g'(v) g'(0)} \frac { \begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 1 & e_1 & e_1^2 & 0 \\ \end{vmatrix} \space \begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 1 & e_2 & e_2^2 & 0 \\ \end{vmatrix} \space \begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 1 & e_3 & e_3^2 & 0 \\ \end{vmatrix} \space \begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 1 & e_4 & e_4^2 & 0 \\ \end{vmatrix} } {\begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 0 & 0 & 1 & \sqrt{a} \\ \end{vmatrix}^2 \space \begin{vmatrix} 1 & g(u) & g(u)^2 & g'(u) \\ 1 & g(v) & g(v)^2 & g'(v) \\ 1 & g(0) & g(0)^2 & -g'(0) \\ 0 & 0 & 1 & -\sqrt{a} \\ \end{vmatrix}^2} \end{equation} where $e_1,e_2,e_3,e_4$ are the four roots of the right hand side of \eqref{eq:curve}.
The "Abel Sum" \begin{equation} \frac {dx_1} {\sqrt{(x_1-e_1)(x_1-e_2)(x_1-e_3)(x_1-e_4)}} \enspace + \enspace \frac {dx_2} {\sqrt{(x_2-e_1)(x_2-e_2)(x_2-e_3)(x_2-e_4)}} \enspace + \enspace \frac {dx_3} {\sqrt{(x_3-e_1)(x_3-e_2)(x_3-e_3)(x_3-e_4)}} \enspace = \enspace 0 \end{equation} has an algebraic integral given by \begin{equation} \begin{vmatrix} \sqrt{x_1 - e_1} & \sqrt{x_1 - e_2} & \sqrt{x_1 - e_3} & \sqrt{x_1 - e_4} \\ \sqrt{x_2 - e_1} & \sqrt{x_2 - e_2} & \sqrt{x_2 - e_3} & \sqrt{x_2 - e_4} \\ \sqrt{x_3 - e_1} & \sqrt{x_3 - e_2} & \sqrt{x_3 - e_3} & \sqrt{x_3 - e_4} \\ \sqrt{x_4 - e_1} & \sqrt{x_4 - e_2} & \sqrt{x_4 - e_3} & \sqrt{x_4 - e_4} \\ \end{vmatrix} \enspace = \enspace 0 \end{equation} where $x_4$ is the constant of integration.