J-invariant

From Elliptic Curve Crypto
Revision as of 16:58, 3 January 2025 by Rational Point (talk | contribs) (bad numbers and no refs)

The j-invariant 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^2+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 j=\frac{6912a^3}{4a^3+27b^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 j=\frac{-110592a^3}{\Delta}}

in terms of the elliptic discriminant Δ [1].

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