2.6. Elliptic curves over finite fields 37

equation f(x, y) = 0, where

f(x, y) =

y2

+ a1xy + a3y −

x3

−

a2x2

− 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 −

x0)2

+ λ2(x − x0)(y − y0) + λ3(y −

y0)2

− (x −

x0)3

= ((y − y0) − α(x − x0)) · ((y − y0) − β(x − x0)) − (x −

x0)3

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

unstable) reduction.

(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 :

y2

=

x3

+35x+5 has good reduction at p = 7, because

y2

≡

x3

+ 5 mod 7 is a non-singular curve over F7.