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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 differentiated 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} \label{eq:simple_cubic_curv} y^2 \enspace = \enspace ax^3 \space + \space bx^2 \space + cx \space + \space d \end{equation}
• Notable reduced form $\enspace y^2 = 4x^3 - g_2 x - g_3\enspace $
• Monomial basis sequence $\enspace 1,x, y, x^2, xy, x^2y \enspace \ldots$
• Interpolating curves are straight lines

Simple Quartic Curve

• Differential equation for a general order 2 elliptic function \begin{equation} \label{eq:simple_quartic_curve} y^2 \enspace = \enspace ax^4 \space + \space bx^3 \space + cx^2 \space + \space dx \space + \space e \end{equation}
• Invariant under differentiated Möbius transformations
• Monomial basis sequence $1, x, x^2, y, \quad x^3, xy, \quad x^4, x^2y \quad \ldots$
• Interpolating curves are parabola's

Scwharz-Christoffel 3-Symmetry Curve

• Differential equation for an order 3 elliptic function with three-fold symmetry \begin{equation} \label{eq:sc_curve_3} y^3 \enspace = \enspace \left(ax^3 \space + \space bx^2 \space + \space cx \space + \space d \right)^2 \end{equation}
• Invariant under differentiated Möbius transformations

Scwharz-Christoffel 4-Symmetry Curve

• Differential equation for an order 4 elliptic function with four-fold symmetry \begin{equation} \label{eq:sc_curve_4} y^4 \enspace = \enspace \left(ax \space + \space b\right)^2 \left(px^2 \space + \space qx \space + \space r \right)^3 \end{equation}
• Invariant under differentiated Möbius transformations

Scwharz-Christoffel 6-Symmetry Curve

• Differential equation for an order 6 elliptic function with six-fold symmetry \begin{equation} \label{eq:sc_curve_6} y^6 \enspace = \enspace \left(ax \space + \space b\right)^3 \left(cx \space + \space d \right)^4 \left(ex \space + \space f \right)^5 \end{equation}
• Invariant under differentiated Möbius transformations

Sextic Curve

• Differential equation for a general order 3 elliptic function \begin{equation} \label{eq:sextic_curve} y^3 \enspace + \enspace \left(ax^2 \space + \space bx \space + \space c \right)y^2 \enspace + \enspace \left(px^3 \space + \space qx^2 \space + \space rx \space + \space s \right)^2 \enspace - \enspace \tfrac {4} {27} \left(ax^2 \space + \space bx \space + \space c \right)^3 \enspace = \enspace 0 \end{equation}
• Invariant under differentiated Möbius transformations

Cubic Curve

• Algebraic relation between two order 3 elliptic functions with the same poles \begin{equation} \label{eq:cubic_curve} py^3 \enspace + \enspace \left(qx \space + \space r\right)y^2 \enspace + \enspace \left(sx^2 \space + \space tx \space + \space u\right)y \enspace + \enspace \left(a x^3 \space + \space b x^2 \space + \space c x \space + \space d\right) \enspace = \enspace 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 $ \enspace 1, x, y, \quad x^2, xy, y^2, \quad x^3, x^2y, xy^2, \quad x^4, x^3y, x^2y^2 \enspace \ldots$
• Interpolating curves are straight lines

Biquadratic Curve

• Algebraic relation between two general order 2 elliptic functions \begin{equation} \label{eq:biquadratic_curve} \left(px^2 \space + \space qx \space + \space r\right)y^2 \enspace + \enspace \left(sx^2 \space + \space tx \space + \space u\right)y \enspace + \enspace \left(ax^2 \space + \space bx \space + \space c\right) \enspace = \enspace 0 \end{equation}
• Invariant under Möbius transformations.
• Monomial basis sequence $ \enspace 1, x, y, xy, \quad x^2, x^2y, xy^2, y^2, \quad x^3,x^3y,xy^3,y^3, \quad x^4,x^4y,xy^4,y^4 \enspace \ldots$
• Interpolating curves are hyperbola's

General 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} \label{eq:quartic_curve} hy^2 \enspace + \enspace \left(px^2 \space + \space qx \space + \space r\right)y \enspace + \enspace \left(a x^4 \space + \space b x^3 \space + \space c x^2 \space + \space d x \space + \space e\right) \enspace = \enspace 0 \end{equation}
• Invariant under differentiated Möbius transformations.
• Monomial basis sequence $ \enspace 1, x, x^2, y, \quad x^3, xy, \quad x^4, x^2y \enspace \ldots$
• Interpolating curves are parabola's

Legendre Curve

• Differential equation for order 2 elliptic functions which are perfect squares \begin{equation} \label{eq:legendre_curve} y^2 \enspace = \enspace ax^3 \space + \space bx^2 \space + \space cx \end{equation}
• Notable reduced form $ \enspace y^2 = x(x-1)(x-\lambda) \enspace$

Euler Curve

• Differential equation for order 2 elliptic functions with even-odd symmetry \begin{equation} \label{eq:euler_curve} y^2 \enspace = \enspace ax^4 \space + \space bx^2 \space + \space c \end{equation}
• Notable reduced forms $\displaystyle \qquad \begin{eqnarray*} y^2 & = & (1 - x^2)(1 - k^2 x^2) \\[6px] y^2 & = & (x^2 - \alpha^2)(x^2 - \alpha^{-2}) \end{eqnarray*}$

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} \label{eq:edwards_curve} a x^2y^2 \enspace + \enspace b x^2 \enspace + \enspace c y^2 \enspace + \enspace d \enspace = \enspace 0 \end{equation}
• Notable reduced form $ \enspace x^2 + y^2 = 1 + k^2 x^2 y^2 \enspace$

Hofstadter Curve

• Algebraic relation between two order 2 elliptic functions with even-odd symmetry, both with odd symmetry \begin{equation} \label{eq:hofstadter_curve} a x^2y \enspace + \enspace b xy^2 \enspace + \enspace c x \enspace + \enspace d y \enspace = \enspace 0 \end{equation}
• Notable reduced form $ \displaystyle \enspace y + \frac 1 y = \sqrt{\lambda} \left(x + \frac 1 x \right) \enspace$

Hesse Curve

• Algebraic relation between two order 3 elliptic functions with triple symmetry and same poles \begin{equation} \label{eq:hesse_curve} ax^3 \enspace + \enspace by^3 \enspace + \enspace c \enspace + \enspace dxy \enspace = \enspace 0 \end{equation}
• The symmetries in this curve are more obvious when written in homogenous form

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 \eqref{eq: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 \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*} The monomial basis sequence is \begin{equation*} 1, \space x, \space x^2, \space y, \space x^3, \space x^2y \space \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} \enspace = \enspace \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 \space + \space Bx \space + \space Cx^2 \space + \space Dy \enspace = \enspace 0 \end{equation*} The two variable "ratio of resultants" addition formula for $f(z)$ is \begin{equation*} f(u + v) \enspace = \enspace \frac 1 {f(u) f(v) f(0)} \frac {\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 & \sqrt{e} \\ \end{vmatrix} \space \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 & -\sqrt{e} \\ \end{vmatrix}} {\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 & \sqrt{a} \\ \end{vmatrix} \space \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 & -\sqrt{a} \\ \end{vmatrix}} \end{equation*} and for $f'(z)$ it is \begin{equation*} f'(u + v) \enspace = \enspace \frac {a^2} {f'(u) f'(v) f'(0)} \frac { \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_1 & e_1^2 & 0 \\ \end{vmatrix} \space \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_2 & e_2^2 & 0 \\ \end{vmatrix} \space \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_3 & e_3^2 & 0 \\ \end{vmatrix} \space \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_4 & e_4^2 & 0 \\ \end{vmatrix} } {\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 & \sqrt{a} \\ \end{vmatrix}^2 \space \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 & -\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:simple_quartic_curve}.