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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 always 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 Edgar Costa on 2020-11-10 09:08:29
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