J-invariant: Difference between revisions

Latest revision as of 23:26, 25 February 2025

The j-invariant of an elliptic curve in Weierstraß normal form

y^{2}=x^{3}+ax+b

is defined to be

j={\frac {6912a^{3}}{4a^{3}+27b^{2}}}

or

j={\frac {-110592a^{3}}{\Delta }}

in terms of the elliptic discriminant Δ ^[1], or,

j={\frac {64a^{3}}{D}}

in terms of the discriminant D of the cubic polynomial $x^{3}+ax+b$ .

@@ Line 1: / Line 1: @@
 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
-:<math>j=\frac{1728a^3}{4a^3+27b^2}</math>
+:<math>j=\frac{6912a^3}{4a^3+27b^2}</math>
 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>.