Generalization of Vélu’s Formulae for Isogenies between Elliptic Curves

Abstract Given an elliptic curve E and a finite subgroup G, Vélu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P +G as the difference between the abscissa of IG(P ) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P+G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.