Discriminant: Difference between revisions
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two discriminants, not the same |
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The '''discriminant | The '''discriminant of an elliptic curve''' in [[Weierstraß normal form]] | ||
:<math>y^2 = x^3 + ax + b</math> | :<math>y^2 = x^3 + ax + b</math> | ||
| Line 7: | Line 7: | ||
:<math>\Delta = -64a^3-432b^2</math> | :<math>\Delta = -64a^3-432b^2</math> | ||
when the field characteristic is not 2 or 3 <ref>Wolfram MathWorld: Elliptic Discriminant. https://mathworld.wolfram.com/EllipticDiscriminant.html</ref>. | or | ||
:<math>\Delta = \frac{-110592a^3}j</math> | |||
in terms of the [[j-invariant]] when the field characteristic is not 2 or 3 <ref>Wolfram MathWorld: Elliptic Discriminant. https://mathworld.wolfram.com/EllipticDiscriminant.html</ref>. | |||
The '''discriminant of the cubic polynomial''' <ref>Murray R. Spiegel and John Liu. ''Schaum’s Easy Outlines: Mathematical Handbook of Formulas and Tables,'' McGraw–Hill, 2012, pp. 13–15.</ref> in the same form is defined somewhat differently from that of the “elliptic curve” as such, and in fact <math>\Delta = -1728D</math>, where | |||
:<math>D = \frac{4a^3+27b^2}{108}</math> | |||
or | |||
:<math>D = \frac{64a^3}{j}</math> | |||
in terms of the ''j''-invariant. | |||
Latest revision as of 06:47, 8 February 2025
The discriminant of an elliptic curve in Weierstraß normal form
is defined to be
or
in terms of the j-invariant when the field characteristic is not 2 or 3 [1].
The discriminant of the cubic polynomial [2] in the same form is defined somewhat differently from that of the “elliptic curve” as such, and in fact , where
or
in terms of the j-invariant.
- ↑ Wolfram MathWorld: Elliptic Discriminant. https://mathworld.wolfram.com/EllipticDiscriminant.html
- ↑ Murray R. Spiegel and John Liu. Schaum’s Easy Outlines: Mathematical Handbook of Formulas and Tables, McGraw–Hill, 2012, pp. 13–15.
