Discriminant

The discriminant of an elliptic curve in Weierstraß normal form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = x^3 + ax + b}

is defined to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta = -64a^3-432b^2}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta = \frac{-110592a^3}j}

in terms of the j-invariant when the field characteristic is not 2 or 3 .

The discriminant of the cubic polynomial in the same form is defined somewhat differently from that of the “elliptic curve” as such, and in fact Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta =-1728D} , where

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or

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in terms of the j-invariant.