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 $T=-4P$, $S=-{\frac {3}{2}}P$, $R=-{\frac {2}{3}}P$, and $Q=-{\frac {1}{4}}P$.

$\{P\}\longrightarrow \left\{-4P,-{\frac {3}{2}}P,-{\frac {2}{3}}P,-{\frac {1}{4}}P\right\}$

Curve fittings through two points

Choosing points P and Q it is possible to solve for $2P+2Q+T=O$ and $P+Q+3R=O$ by curve-fitting: $T=-2(P+Q)$ and $R=-{\frac {1}{3}}(P+Q)$, if scalar division is permitted. Also $2P+Q+2S=O$ and $P+2Q+2U=O$ so $S=-\left(P+{\frac {1}{2}}Q\right)$ and $U=-\left({\frac {1}{2}}P+Q\right)$, etc.

$\{P,Q\}\longrightarrow \left\{-2(P+Q),-{\frac {1}{3}}(P+Q),-\left(P+{\frac {1}{2}}Q\right),-\left({\frac {1}{2}}P+Q\right),-(3P+Q),-(P+3Q)\right\}$

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.

Point averaging

Point averaging is possible.

${\frac {P+Q}{2}}=-{\frac {1}{4}}{\big (}-2(P+Q){\big )}$.