Legendre elliptic curves over finite fields

Throughout this paper, q > 1 denotes a power of an odd prime number p, and k is a field. Given two elliptic curves E/k and E′/k, all morphisms from E to E′ are understood to be defined over k. In particular, we simply write End(E) for the ring of all endomorphisms of E/k. The notation E ≃ E′ indicates that E is isomorphic to E′, and E ∼ E′ means that E and E′ are isogenous. The endomorphism of multiplication by m ∈ Z on E is denoted by [m]. In case k = Fq, it is a well known fact (see [13]) that E ∼ E′ if and only if |E(Fq)| = |E′(Fq)|. The Frobenius endomorphism on an elliptic curve E/Fq will be denoted by φ = φq. For char(k) 6= 2 and λ ∈ k \ {0, 1}, the Legendre elliptic curve Eλ/k is given by the equation y = x(x − 1)(x − λ). All its 2-torsion points are rational. An arbitrary elliptic curve E/k with this property has an equation of the form y = x(x− α)(x − β) with α, β ∈ k∗ (after a suitable choice of coordinates). Investigating the possible transformations (see [12, III §1]) yields that E is Legendre isomorphic (i.e., isomorphic to a Legendre elliptic curve) if and only if at least one of ±α,±β,±(α− β) is a square in k. This is always true when k( √ −1) is algebraically closed or when k = Fq with q ≡ 3 mod 4, but not, e.g., for k = F13, α = −2 and β = 5. So