Quintic point group operation

Suppose we have a quintic curve in the form

$y^{2}=px^{5}+qx^{4}+rx^{3}+sx^{2}+tx+u$.

Through any four points on a quintic curve in this form, an ordinary elliptic curve in the form

$y^{2}=ax^{3}+bx^{2}+cx+d$

may be fitted, and this elliptic curve must intersect the quintic curve at a fifth point, uniquely determined, up to sign of the y-axis.

The quincunx equations

If P, Q, R, S and T are the five points of intersection between the quintic curve and the fitted elliptic curve, permitting multiplicity, let

$P+Q+R+S+T=O$,

where O is the additional “point at infinity” considered to lie on the curve and serve as an identity for its additive point group operation.

Curve fittings through one point

Choose a point P and solve $4P+T=O$, $3P+2S=O$, and $2P+3R=P+4Q=O$ by curve-fitting. Now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=-4P} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=-\frac32P} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=-\frac23P} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q=-\frac14P} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{P\}\longrightarrow\left\{-4P,-\frac32P,-\frac23P,-\frac14P\right\}}

Curve fittings through two points

Choosing points P and Q it is possible to solve for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2P+2Q+T=O} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P+Q+3R=O} by curve-fitting: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=-2(P+Q)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=-\frac13(P+Q)} , if scalar division is permitted. Also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2P+Q+2S=O} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P+2Q+2U=O} so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=-\left(P+\frac12Q\right)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U=-\left(\frac12P+Q\right)} , etc.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{P,Q\}\longrightarrow\left\{-2(P+Q),-\frac13(P+Q),-\left(P+\frac12Q\right),-\left(\frac12P+Q\right),-\left(P+\frac13Q\right),-\left(\frac13P+Q\right)\right\}}

Goals and objectives

The goal is to reach the point $-P$ from the point P alone, and to reach the point $P+Q$ from the points P and Q if possible, and if so, by the shortest possible path of computation, i.e., using the least number of elliptic curve fittings.