j-Invariant
The j-invariant is an invariant of an elliptic curve under isomorphism over an algebraic closure. For a general Weierstrass equation
| y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, |
(1)
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define
The standard invariants of the equation are
Here, c_4 is the weight-four invariant and Delta is the elliptic discriminant. For Delta!=0, the j-invariant is
| [画像: j(E)=(c_4^3)/Delta. ] |
(8)
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Over a field whose field characteristic is different from 2 and 3, an elliptic curve can be written in short Weierstrass form y^2=x^3+ax+b, in which case
Two elliptic curves over an algebraically closed field are isomorphic if and only if they have the same j-invariant. Over a non-algebraically closed base field, curves with the same j-invariant need not be isomorphic. Such curves may be twists of one another. Over a field whose field characteristic is different from 2 and 3, the exceptional values j=0 and j=1728 correspond to curves having additional automorphisms.
Over C, an elliptic curve is analytically isomorphic to a complex torus C/(Z+tauZ) for tau in the upper half-plane, and its j-invariant is the value j(tau) of the modular j-function. The Fourier expansion and special values of j(tau), together with its role in the theory of complex multiplication, provide further arithmetic information about the j-invariant.
See also
Elliptic Curve, Elliptic Discriminant, Frey Curve, j-Function, Weierstrass FormExplore with Wolfram|Alpha
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References
Silverman, J. H. The Arithmetic of Elliptic Curves, 2nd ed. New York: Springer, 2009.Stepanov, S. A. "The j-Invariant." §7.2 in Codes on Algebraic Curves. New York: Kluwer, pp. 178-180, 1999.Referenced on Wolfram|Alpha
j-InvariantCite this as:
Weisstein, Eric W. "j-Invariant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/j-Invariant.html