The discriminant $\Delta$ of a Weierstrass equation $y^2+h(x)y=f(x)$ can be computed as $$ \Delta := \begin{cases} 2^8\text{lc}(f)^2\text{disc}(f+h^2/4)&\text{if }f+h^2/4\text{ has odd degree},\\ 2^8\text{disc}(f+h^2/4)&\text{if }f+h^2/4\text{ has even degree}, \end{cases} $$ where $\text{lc}(f)$ denotes the leading coefficient of $f$ and $\text{disc}(f)$ its discriminant.

The **discriminant** of a genus 2 curve over $\Q$ is the discriminant of a minimal equation for the curve; it is an invariant of the curve that does not depend on the choice of minimal equation.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2021-01-19 18:23:20

**Referred to by:**

- g2c.1225.a.6125.1.top
- g2c.20736.l.373248.1.bottom
- g2c.363.a.11979.1.top
- g2c.450.a.2700.1.top
- g2c.529.a.529.1.top
- g2c.841.a.841.1.top
- g2c.abs_discriminant
- g2c.conductor
- g2c.g2curve
- g2c.invariants
- g2c.local_invariants
- g2c.minimal_equation
- rcs.cande.g2c
- rcs.source.g2c
- lmfdb/genus2_curves/templates/g2c_curve.html (line 58)
- lmfdb/genus2_curves/web_g2c.py (line 661)

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

- 2021-01-19 18:23:20 by Andrew Sutherland (Reviewed)
- 2018-05-23 16:37:04 by John Cremona (Reviewed)

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