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).
- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-01-06 14:44:06
- 2020-01-06 14:44:06 by Andrew Sutherland (Reviewed)
- 2020-01-06 14:39:25 by Andrew Sutherland
- 2020-01-06 14:39:15 by Andrew Sutherland
- 2020-01-06 14:38:54 by Andrew Sutherland