As we know, an elliptic curve defined over Fq with a rational 2-torsion subgroup can be expressed in the special form (up to twists). Accordingly a natural question arises about the number of distinct (up to isomorphism) elliptic curves over Fq in the family.
Dear Davood,
Here are links and attached files in subject.
-On the number of distinct elliptic curves in some families - ResearchGate
https://www.researchgate.net/.../220639009_On_the_number_of_d...
-On the number of distinct elliptic curves in some families | SpringerLink
link.springer.com/article/10.1007/s10623-009-9310-2
-On the isomorphism classes of Legendre elliptic curves ... - Springer
link.springer.com/content/.../10.1007%2Fs11425-011-4255-0.pdf
Best regards
Dear Davood,
Computing the cardinalities of the groups of rational points on elliptic curves over finite fields is a fundamental algorithmic challenge in computational number theory, and an essential tool in elliptic curve cryptography.
https://arxiv.org/pdf/1605.07749.pdf
https://arxiv.org/pdf/1001.2871.pdf
https://arxiv.org/pdf/1510.01849.pdf
https://link.springer.com/article/10.1007/s10623-009-9310-2
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