Every (smooth, projective, geometrically integral) curve of genus 2 can be defined by a **Weierstrass equation** of the form $$y^2+h(x)y=f(x)$$
with nonzero discriminant and $\deg h \le 3$ and $\deg f \le 6$; in order to have genus 2 we must have $\deg h = 3$ or $\deg f =5,6$. Over a field whose characteristic is not 2 one can complete the square to make $h(x)$ zero, but this will yield a model with bad reduction at 2 that is typically not a minimal equation for the curve.

This equation can be viewed as defining the function field of the curve, or as a smooth model of the curve in the weighted projective plane. Every curve of genus 2 admits a degree 2 cover of the projective line (consider the function $x$) and is therefore a hyperelliptic curve.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-25 13:56:16

**Referred to by:**

- g2c.abs_discriminant
- g2c.aut_grp
- g2c.g2_invariants
- g2c.igusa_clebsch_invariants
- g2c.igusa_invariants
- g2c.invariants
- g2c.jacobian
- g2c.label
- g2c.local_invariants
- g2c.mw_generator
- g2c.regulator
- g2c.search_input
- g2c.st_group
- rcs.rigor.lfunction.curve
- rcs.source.g2c
- lmfdb/genus2_curves/main.py (line 421)

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

- 2020-10-25 13:56:16 by Andrew Sutherland (Reviewed)
- 2020-10-25 13:55:23 by Andrew Sutherland
- 2018-05-23 16:46:47 by John Cremona (Reviewed)

**Differences**(show/hide)