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
  y^2 = ax^3 + bx^2 + cx + d
• Notable reduced form
  y^2 = 4x^3 - g_2 x - g_3

Simple Quartic Curve

• Differential equation for a general order 2 elliptic function
  y^2 = ax^4 + bx^3 + cx^2 + dx + e
• Invariant under Möbius transformations

Scwharz-Christoffel 3-Symmetry Curve

• Differential equation for an order 3 elliptic function with three-fold symmetry
  y^3 = \left(ax^3 + bx^2 + cx + d \right)^2

• Invariant under Möbius transformations

Scwharz-Christoffel 4-Symmetry Curve

• Differential equation for an order 4 elliptic function with four-fold symmetry
  y^4 = \left(ax + b\right)^2 \left(px^2 + qx + r \right)^3

• Invariant under Möbius transformations

Scwharz-Christoffel 6-Symmetry Curve

• Differential equation for an order 6 elliptic function with six-fold symmetry
  y^6 = \left(ax + b\right)^3 \left(cx + d \right)^4 \left(ex + f \right)^5

• Invariant under Möbius transformations

Sextic Curve

• Differential equation for a general order 3 elliptic function
  y^3 + \left(ax^2 + bx + c \right)y^2 + \left(px^3 + qx^2 + rx + s \right)^2 - \tfrac {4} {27} \left(ax^2 + bx + c \right)^3 = 0

• Invariant under Möbius transformations

Cubic Curve

• Algebraic relation between two order 3 elliptic functions with the same poles
  py^3 + \left(qx + r\right)y^2 + \left(sx^2 + tx + u\right)y + \left(a x^3 + b x^2 + c x + d\right) = 0

• 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
  \left(px^2 + qx + r\right)y^2 + \left(sx^2 + tx + u\right)y + \left(ax^2 + bx + c\right) = 0

• 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.
  hy^2 + \left(px^2 + qx + r\right)y + \left(a x^4 + b x^3 + c x^2 + d x + e\right) = 0

• 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
  y^2 = ax^3 + bx^2 + cx

• Notable reduced form
  y^2 = x(x-1)(x-\lambda)

Euler Curve

• Differential equation for order 2 elliptic functions with even-odd symmetry
  y^2 = ax^4 + bx^2 + c

• Notable reduced forms
  y^2 = (1 - x^2)(1 - k^2 x^2)

  and

  y^2 = (x^2 - \alpha^2)(x^2 - \alpha^{-2})

Edwards Curve

• Algebraic relation between two order 2 elliptic functions with even-odd symmetry, one with even symmetry and the other with odd symmetry
  a x^2y^2 + bx^2 + cy^2 + d = 0

• Notable reduced form
  x^2 + y^2 = 1 + k^2 x^2 y^2

Hofstadter Curve

• Algebraic relation between two order 2 elliptic functions with even-odd symmetry, both with odd symmetry
  ax^2y + bxy^2 + cx + dy = 0

• Notable reduced form
  y + \frac 1 y = \sqrt{\lambda} \left(x + \frac 1 x \right)

Hesse Curve

• Algebraic relation between two order 3 elliptic functions with triple symmetry and same poles
  ax^3 + by^3 + c + dxy = 0

• The symmetries in this curve are more obvious when written in homogenous form
  ax^3 + by^3 + cz^3 + dxyz = 0

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

The monomial basis sequence is