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The discriminant $\Delta$ of a Weierstrass equation $y^2+h(x)y=f(x)$ can be computed as $$ \Delta := \begin{cases} 2^8\text{lc}(f)^2\text{disc}(f+h^2/4)&\text{if }f+h^2/4\text{ has odd degree},\\ 2^{10}\text{disc}(f+h^2/4)&\text{if }f+h^2/4\text{ has even degree}, \end{cases} $$ 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.

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  • Last edited by John Cremona on 2018-05-23 16:37:04
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