On Various Families of Twisted Jacobi Quartics
We provide several results on some families of twisted Jacobi quartics. We give new addition formulæ for two models of twisted Jacobi quartic elliptic curves, which represent respectively 1/6 and 2/3 of all elliptic curves, with respective costs 7M + 3S + D a and 8M + 3 S + D a . These formulæ allow addition and doubling of points, except for points differing by a point of order two.
Furthermore, we give an intrinsic characterization of elliptic curves represented by the classical Jacobi quartic, by the action of the Frobenius endomorphism on the 4-torsion subgroup. This allows us to compute the exact proportion of elliptic curves representable by various models (the three families of Jacobi quartics, plus Edwards and Huff curves) from statistics on this Frobenius action.
KeywordsConjugacy Class Elliptic Curve Elliptic Curf Asymptotic Probability Intrinsic Characterization
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