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).

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**Knowl status:**

- Review status: reviewed
- Last edited by Edgar Costa on 2020-11-10 09:08:29

**History:**(expand/hide all)

- 2020-11-10 09:08:29 by Edgar Costa (Reviewed)
- 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

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