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^2+ax+b</math>
:<math>y^2=x^3+ax+b</math>


is defined to be
is defined to be


:<math>j=\frac{1728a^3}{4a^3+27b^2}</math>
:<math>j=\frac{6912a^3}{4a^3+27b^2}</math>


or
or


:<math>j=\frac{-108a^3}{\Delta}</math>
:<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 .

  1. Wolfram MathWorld: j-Invariant. https://mathworld.wolfram.com/j-Invariant.html