36 2. Elliptic curves

Example 2.6.2. Sometimes the reduction of a model for an elliptic

curve E modulo a prime p is not smooth, but it is smooth for some

other models of E. For instance, consider the curve E :

y2

=

x3

+

15625. Then E ≡ E mod 5 is not smooth over F5 because the point

(0,0) mod 5 is a singular point. However, using the invertible change

of variables (x, y) →

(52X,53Y

), we obtain a new model over Q for

E given by E : Y

2

=

X3

+ 1, which is smooth when we reduce it

modulo 5. The problem here is that the model we chose for E is not

minimal. We describe what we mean by minimal next.

Definition 2.6.3. Let E be an elliptic curve given by y2 = x3 +Ax+

B, with A, B ∈ Q.

(1) We define ΔE, the discriminant of E, by

ΔE =

−16(4A3

+

27B2).

For a definition of the discriminant for more general Weier-

strass equations, see for example [Sil86], p. 46.

(2) Let S be the set of all elliptic curves E that are isomor-

phic to E over Q (see Definition 2.2.4) and such that the

discriminant of E is an integer. The minimal discriminant

of E is the integer ΔE that attains the minimum of the set

{|ΔE | : E ∈ S}. In other words, the minimal discriminant

is the smallest integral discriminant (in absolute value) of

an elliptic curve that is isomorphic to E over Q. If E is the

model for E with minimal discriminant, we say that E is a

minimal model for E.

Example 2.6.4. The curve E :

y2

=

x3

+

56

has discriminant

ΔE =

−2433512,

and the curve E :

y2

=

x3

+ 1 has discriminant

ΔE =

−2433.

Since E and E are isomorphic (see Definition 2.2.4

and Example 2.6.2), then ΔE cannot be the minimal discriminant for

E and

y2

=

x3

+

56

is not a minimal model. In fact, the minimal

discriminant is ΔE = −432 and E is a minimal model.

Before we go on to describe the types of reduction modulo p that

one can encounter, we need a little bit of background on types of

singularities. Let E be a cubic curve over a field K with Weierstrass