List Of Curves

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In this section we compute addition formulae for cubic, biquadratic and quartic elliptic curves based on monomial basis sequences.

Some points to note

This list is not exhaustive, for example some curves which arise from simple inversions $x \rightarrow 1/x$ of listed curves are not included.
Invariant under Möbius transformations means the family as a whole is invariant under the birational transformation $(x,y) \rightarrow(L(x),yL'(x))$ where $L$ is a Möbius transformation.
Invariance under Möbius transformations implies the $g_2, g_3$ and $j$ invariants of the curve can be expressed in terms of the binary invariants of the coefficient polynomials.

Simple Cubic Curve

Differential equation for an order 2 elliptic function with a double pole
\begin{equation*} y^2 \sp = \sp ax^3 \sp + \sp bx^2 \sp + \sp cx \sp + \sp d \end{equation*}
Notable reduced form
\begin{equation*} y^2 \sp = \sp 4x^3 \sp - \sp g_2 x \sp - \sp g_3 \end{equation*}

Simple Quartic Curve

Differential equation for a general order 2 elliptic function
\begin{equation*} y^2 \sp = \sp ax^4 \sp + \sp bx^3 \sp + \sp cx^2 \sp + \sp dx \sp + \sp e \end{equation*}
Invariant under Möbius transformations

Scwharz-Christoffel 3-Symmetry Curve

Differential equation for an order 3 elliptic function with three-fold symmetry

\begin{equation*} y^3 \sp = \sp \left(ax^3 + bx^2 + cx + d \right)^2 \end{equation*}
Invariant under Möbius transformations

Scwharz-Christoffel 4-Symmetry Curve

Differential equation for an order 4 elliptic function with four-fold symmetry

\begin{equation*} y^4 \sp = \sp \left(ax + b\right)^2 \left(px^2 + qx + r \right)^3 \end{equation*}
Invariant under Möbius transformations

Scwharz-Christoffel 6-Symmetry Curve

Differential equation for an order 6 elliptic function with six-fold symmetry

\begin{equation*} y^6 \sp = \sp \left(ax + b\right)^3 \left(cx + d \right)^4 \left(ex + f \right)^5 \end{equation*}
Invariant under Möbius transformations

Sextic Curve

Differential equation for a general order 3 elliptic function

\begin{equation*} y^3 \sp + \sp \left(ax^2 + bx + c \right)y^2 \sp + \sp \left(px^3 + qx^2 + rx + s \right)^2 \sp - \sp \tfrac {4} {27} \left(ax^2 + bx + c \right)^3 \sp = \sp 0 \end{equation*}
Invariant under Möbius transformations

Cubic Curve

Algebraic relation between two order 3 elliptic functions with the same poles

\begin{equation*} py^3 \sp + \sp \left(qx + r\right)y^2 \sp + \sp \left(sx^2 + tx + u\right)y \sp + \sp \left(a x^3 + b x^2 + c x + d\right) \sp = \sp 0 \end{equation*}
Invariant under joint linear fractional transformations - and therefore the $g_2$ and $g_3$ invariants can be expressed in terms of the ternary invariants of the homogenised equation
Monomial basis sequence starts with $1, x, y$ implying the interpolating curves are straight lines

Biquadratic Curve

Algebraic relation between two general order 2 elliptic functions

\begin{equation*} \left(px^2 + qx + r\right)y^2 \sp + \sp \left(sx^2 + tx + u\right)y \sp + \sp \left(ax^2 + bx + c\right) \sp = \sp 0 \end{equation*}
Invariant under Möbius transformations.
Monomial basis sequence starts with $1, x, y, xy$ implying the interpolating curves are hyperbola's

Quartic Curve

Algebraic relation between a general order 2 elliptic function with two single poles and an order 4 elliptic function with two double poles at same location.

\begin{equation*} hy^2 \sp + \sp \left(px^2 + qx + r\right)y \sp + \sp \left(a x^4 + b x^3 + c x^2 + d x + e\right) \sp = \sp 0 \end{equation*}
Invariant under Möbius transformations.
Monomial basis sequence starts with $1, x, x^2, y$ implying the interpolating curves are parabola's

Legendre Curve

Differential equation for order 2 elliptic functions which are perfect squares

\begin{equation*} y^2 \sp = \sp ax^3 \sp + \sp bx^2 \sp + \sp cx \end{equation*}
Notable reduced form

\begin{equation*} y^2 \sp = \sp x(x-1)(x-\lambda) \end{equation*}

Euler Curve

Differential equation for order 2 elliptic functions with even-odd symmetry

\begin{equation*} y^2 \sp = \sp ax^4 \sp + \sp bx^2 \sp + \sp c \end{equation*}
Notable reduced forms

\begin{equation*} y^2 \sp = \sp (1 - x^2)(1 - k^2 x^2) \end{equation*}

and

\begin{equation*} y^2 \sp = \sp (x^2 - \alpha^2)(x^2 - \alpha^{-2}) \end{equation*}

Edwards Curve

Algebraic relation between two order 2 elliptic functions with even-odd symmetry, one with even symmetry and the other with odd symmetry

\begin{equation*} a x^2y^2 \sp + \sp bx^2 \sp + \sp cy^2 \sp + \sp d \sp = \sp 0 \end{equation*}
Notable reduced form

\begin{equation*} x^2 \sp + \sp y^2 \sp = \sp 1 \sp + \sp k^2 x^2 y^2 \end{equation*}

Hofstadter Curve

Algebraic relation between two order 2 elliptic functions with even-odd symmetry, both with odd symmetry

\begin{equation*} ax^2y \sp + \sp bxy^2 \sp + \sp cx \sp + \sp dy \sp = \sp 0 \end{equation*}
Notable reduced form

\begin{equation*} y + \frac 1 y \sp = \sp \sqrt{\lambda} \left(x + \frac 1 x \right) \end{equation*}

Hesse Curve

Algebraic relation between two order 3 elliptic functions with triple symmetry and same poles

\begin{equation*} ax^3 + by^3 + c + dxy \sp = \sp 0 \end{equation*}
The symmetries in this curve are more obvious when written in homogenous form

\begin{equation*} ax^3 + by^3 + cz^3 + dxyz \sp = \sp 0 \end{equation*}

Monomial Basis Sequence

Every elliptic curve is uniformised by a pair of elliptic functions say $f$ and $g$. A monomial basis sequence is a sequence of monomials in $f$ and $g$ which span the vector space of elliptic functions with the same poles as $f$ and $g$ up to a certain order $n$. If the family of curves has a natural sequence of monomial bases then

Each basis implies a Frobenius-Stickelberger style symmetric addition formula.
It also implies an Abelian style family of interpolating curves.
Elimination between an elliptic curve and an interpolating curve yields an explicit addition formula.
This formula is roughly speaking the ratio of two resultants.

For example the curve simple_quartic_curve. The pair of uniformising elliptic functions are $f$ and $f'$ where $f(z)$ is any non-trivial solution of the differential equation

\begin{equation*} f'(z)^2 \sp = \sp a f(z)^4 + b f(z)^3 + c f(z)^2 + d f(z) + e \end{equation*}

The monomial basis sequence is

\begin{equation*} 1, x, x^2, y, x^3, x^2y \ldots \end{equation*}

The first non-trivial Frobenius-Stickelberger style symmetric addition formula is

\begin{equation*} \begin{vmatrix} 1 & f(z_1) & f(z_1)^2 & f'(z_1) \\ 1 & f(z_2) & f(z_2)^2 & f'(z_2) \\ 1 & f(z_3) & f(z_3)^2 & f'(z_3) \\ 1 & f(z_4) & f(z_4)^2 & f'(z_4) \\ \end{vmatrix} \sp = \sp \frac {2 \thinspace \sigma^4(\rho_1 - \rho_2)} {a^2} \cdot \frac {\sigma\big(z_1 + z_2 + z_3 + z_4 - 2(\rho_1 + \rho_2)\big) \prod\limits_{i \lt j}\sigma(z_i - z_j)} {\prod\limits_{i=1}^4 \sigma^2(z_i - \rho_1) \sigma^2(z_i - \rho_2)} \end{equation*}

where $\rho_1$ and $\rho_2$ are the poles of $f$.

The interpolating curves are parabola's of the form

\begin{equation*} A + Bx + Cx^2 + Dy \sp = \sp 0 \end{equation*}

The two variable "ratio of resultants" addition formula for $f(z)$ is

\begin{equation*} f(u + v) \sp = \sp \frac 1 {f(u) f(v) f(0)} \frac { \prod\limits_{\textsf{signs}} \begin{vmatrix} 1 & f(u) & f(u)^2 & f'(u) \\ 1 & f(v) & f(v)^2 & f'(v) \\ 1 & f(0) & f(0)^2 & -f'(0) \\ 1 & 0 & 0 & \pm\sqrt{e} \\ \end{vmatrix}} {\prod\limits_{\textsf{signs}} \begin{vmatrix} 1 & f(u) & f(u)^2 & f'(u) \\ 1 & f(v) & f(v)^2 & f'(v) \\ 1 & f(0) & f(0)^2 & -f'(0) \\ 0 & 0 & 1 & \pm\sqrt{a} \\ \end{vmatrix}} \end{equation*}

and for $f'(z)$ it is

\begin{equation*} f'(u + v) \sp = \sp \frac {a^2} {f'(u) f'(v) f'(0)} \frac { \prod\limits_{i=1}^4 \begin{vmatrix} 1 & f(u) & f(u)^2 & f'(u) \\ 1 & f(v) & f(v)^2 & f'(v) \\ 1 & f(0) & f(0)^2 & -f'(0) \\ 1 & e_i & e_i^2 & 0 \\ \end{vmatrix}} { \prod\limits_{\textsf{signs}} \begin{vmatrix} 1 & f(u) & f(u)^2 & f'(u) \\ 1 & f(v) & f(v)^2 & f'(v) \\ 1 & f(0) & f(0)^2 & -f'(0) \\ 0 & 0 & 1 & \pm\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 simple_quartic_curve.