2.6. Elliptic curves over finite fields 37
equation f(x, y) = 0, where
f(x, y) =
+ a1xy + a3y −
− a4x − a6,
and suppose that E has a singular point P = (x0,y0), i.e., ∂f/∂x(P ) =
∂f/∂y(P ) = 0. Thus, we can write the Taylor expansion of f(x, y)
around (x0,y0) as follows:
f(x, y) − f(x0,y0)
= λ1(x −
+ λ2(x − x0)(y − y0) + λ3(y −
− (x −
= ((y − y0) − α(x − x0)) · ((y − y0) − β(x − x0)) − (x −
for some λi ∈ K and α, β ∈ K (an algebraic closure of K).
Definition 2.6.5. The singular point P ∈ E is a node if α = β. In
this case there are two different tangent lines to E at P , namely
y − y0 = α(x − x0), y − y0 = β(x − x0).
If α = β, then we say that P is a cusp, and there is a unique tangent
line at P .
Definition 2.6.6. Let E/Q be an elliptic curve given by a minimal
model, let p ≥ 2 be a prime and let E be the reduction curve of E
modulo p. We say that E/Q has good reduction modulo p if E is
smooth and hence is an elliptic curve over Fp. If E is singular at a
point P ∈ E(Fp), then we say that E/Q has bad reduction at p and
we distinguish two cases:
(1) If E has a cusp at P , then we say that E has additive (or
(2) If E has a node at P , then we say that E has multiplicative
(or semistable) reduction. If the slopes of the tangent lines
(α and β as above) are in Fp, then the reduction is said to
be split multiplicative (and non-split otherwise).
Example 2.6.7. Let us see some examples of elliptic curves with
different types of reduction:
(1) E1 :
+35x+5 has good reduction at p = 7, because
+ 5 mod 7 is a non-singular curve over F7.