Displaying similar documents to “Constructing elliptic curves over finite fields using double eta-quotients”

Counting points on elliptic curves over finite fields

René Schoof (1995)

Journal de théorie des nombres de Bordeaux

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We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large ; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic...

Computing modular degrees using L -functions

Christophe Delaunay (2003)

Journal de théorie des nombres de Bordeaux

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We give an algorithm to compute the modular degree of an elliptic curve defined over ℚ . Our method is based on the computation of the special value at s = 2 of the symmetric square of the L -function attached to the elliptic curve. This method is quite efficient and easy to implement.

Good reduction of elliptic curves over imaginary quadratic fields

Masanari Kida (2001)

Journal de théorie des nombres de Bordeaux

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We prove that the j -invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.

Involutory elliptic curves over 𝔽 q ( T )

Andreas Schweizer (1998)

Journal de théorie des nombres de Bordeaux

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For n ∈ 𝔽 q [ T ] let G be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve X 0 ( 𝔫 ) . We determine all 𝔫 and G for which the quotient curve G ∖ X 0 ( 𝔫 ) is rational or elliptic.