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
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