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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
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This list is not exhaustive, for example some curves which arise from simple inversions $x \rightarrow 1/x$ of listed curves are not included.
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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.
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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
Simple Quartic Curve
Scwharz-Christoffel 3-Symmetry Curve
Scwharz-Christoffel 4-Symmetry Curve
Scwharz-Christoffel 6-Symmetry Curve
Sextic Curve
Cubic Curve
Biquadratic Curve
Quartic Curve
Legendre Curve
Euler Curve
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Differential equation for order 2 elliptic functions with even-odd symmetry
y^2 \sp = \sp ax^4 \sp + \sp bx^2 \sp + \sp c
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Notable reduced forms
y^2 \sp = \sp (1 - x^2)(1 - k^2 x^2)
and
y^2 \sp = \sp (x^2 - \alpha^2)(x^2 - \alpha^{-2})
Edwards Curve
Hofstadter Curve
Hesse Curve
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
f'(z)^2 \sp = \sp a f(z)^4 + b f(z)^3 + c f(z)^2 + d f(z) + e
The monomial basis sequence is
1, x, x^2, y, x^3, x^2y \ldots