J-invariant: Difference between revisions
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The ''' ''j''-invariant''' of an elliptic curve in [[Weierstraß normal form]] | The ''' ''j''-invariant''' of an elliptic curve in [[Weierstraß normal form]] | ||
:<math>y^2=x^ | :<math>y^2=x^3+ax+b</math> | ||
is defined to be | is defined to be | ||
:<math>j=\frac{ | :<math>j=\frac{6912a^3}{4a^3+27b^2}</math> | ||
or | or | ||
:<math>j=\frac{- | :<math>j=\frac{-110592a^3}{\Delta}</math> | ||
in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>. | in terms of the elliptic [[discriminant]] Δ <ref>Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html</ref>, or, | ||
:<math>j=\frac{64a^3}{D}</math> | |||
in terms of the discriminant ''D'' of the cubic polynomial <math>x^3+ax+b</math>. |
Latest revision as of 23:26, 25 February 2025
The j-invariant of an elliptic curve in Weierstraß normal form
is defined to be
or
in terms of the elliptic discriminant Δ [1], or,
in terms of the discriminant D of the cubic polynomial .
- ↑ Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html