show · g2c.simple_equation all knowls · up · search:

A hyperelliptic curve over $\Q$ with a minimal equation of the form \[ y^2 + h(x)y = f(x), \] with $h,f\in \Z[x]$ can always be defined by a simplified equation of the form \[ y^2 = g(x), \] with $g\in \Z[x]$ defined by $g:=4f+h^2$.

In the LMFDB, invariants of hyperelliptic curves such as rational points and generators of the Mordell-Weil group of its Jacobian are alwyas expressed in terms of the minimal equation, not the corresponding simplified equation (except in cases where the minimal equation has $h=0$ and the two coincide).

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2020-01-06 14:44:06
Referred to by:
History: (expand/hide all) Differences (show/hide)