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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.