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100 Chapter 3. Bent functions and algebraic curvesa quantity called its discriminant 5 In fact, a Weierstraß equation describes the affine part of asmooth, or non-singular, projective curve — which is then an elliptic curve — if and only if itsdiscriminant is non-zero [244, Proposition III.1.4]. The affine parts of such curves defined overthe real numbers, with discriminants zero and non-zero, are depicted in Figures 3.1, 3.2, 3.3, 3.4,and 3.5.Figure 3.1: The elliptic curve E : y 2 = x 3 + 1, ∆ = −432Over an algebraically closed field, elliptic curves are classified up to isomorphism by thej-invariant, which can be defined as a rational function of the coefficients of the curve 6 [244,Proposition III.1.4]. Furthermore, for any j 0 in K, the algebraic closure of K, there is a curvedefined over K(j 0 ) with j-invariant j 0 [244, Proposition III.1.4].For a given extension L of K, a point is said to be rational if its coordinates lie in L (andnot in a larger extension). There is a composition law, usually additively denoted, on the set ofrational points giving it a group structure [244, Section III.2]. It can be explicitly described bythe so-called “chord-and-tangent” law and is depicted in Figure 3.6. The point at infinity O E isthe neutral element for the addition law. Multiplication by an integer n on E, that we denote by[n], can then be naturally defined.The following result shows that any rational map between curves is the composition of atranslation and a homomorphism.Definition 3.2.2 (Isogeny). Let E 1 and E 2 be two elliptic curves defined over K and φ amorphism between them. We say that φ is an isogeny if φ(O E1 ) = O E2 .5 The expression of the discriminant using the coefficients a 1 , a 3 , a 2 , a 4 and a 6 is quite complicated and we willnot give it here. A simpler expression exists in characteristic different from 2 and 3 and is given in Definition 5.2.1.6 The expression of the j-invariant using the coefficients a 1 , a 3 , a 2 , a 4 and a 6 is also quite complicated andwill not be given here. A simpler expression exists as well in characteristic different from 2 and 3 and is given inDefinition 5.2.2.