2.6. Elliptic curves over finite fields 39

proposition is the discriminant of the polynomial f(x). The discrim-

inant of E/Q, ΔE as in Definition 2.6.3, is a multiple of D; in fact,

ΔE = 16D. This fact together with Proposition 2.6.8 yield the fol-

lowing corollary:

Corollary 2.6.9. Let E/Q be an elliptic curve with coeﬃcients in

Z. Let p ≥ 2 be a prime. If E has bad reduction at p, then p | ΔE.

In fact, if E is given by a minimal model, then p | ΔE if and only if

E has bad reduction at p.

Example 2.6.10. The discriminant of the elliptic curve E1 : y2 =

x3 + 35x + 5 of Example 2.6.7 is ΔE1 = −2754800 = −24 · 52 · 71 · 97

(and, in fact, this is the minimal discriminant of E1). Thus, E1 has

good reduction at 7 but it has bad reduction at 2, 5, 71 and 97. The

reduction at 71 and 97 is multiplicative.

Let E be an elliptic curve defined over a finite field Fq with q

elements, where q = pr and p ≥ 2 is prime. Notice that E(Fq) ⊆

P2(Fq), and the projective plane over Fq only has a finite number

of points (how many?). Thus, the number Nq := |E(Fq)|, i.e., the

number of points on E over Fq, is finite. The following theorem

provides a bound for Nq. This result was conjectured by Emil Artin

(in his thesis) and was proved by Helmut Hasse in the 1930’s:

Theorem 2.6.11 (Hasse; [Sil86], Ch. V, Theorem 1.1). Let E be

an elliptic curve defined over Fq. Then

q + 1 − 2

√

q Nq q + 1 + 2

√

q,

where Nq = |E(Fq)|.

Remark 2.6.12. Heuristically, we expect that Nq is approximately

q+1, in agreement with Hasse’s bound. Indeed, let E/Q be an elliptic

curve given by y2 = x3 + Ax + B, with A, B ∈ Z, and let q = p be a

prime for simplicity. There are p choices of x in Fp. For each value

x0, the polynomial f(x) =

x3

+Ax+B gives a value f(x0) ∈ Fp. The

probability that a random element in Fp is a perfect square in Fp is

1/2 (notice, however, that f(x0) is not random; this is just a heuristic

argument). If f(x0) is a square modulo p, i.e., if there is a y0 ∈ Fp

such that f(x0) ≡ y0

2

mod p, then there are two points (x0, ±y0) in