Elliptic curves of conductor 11

We determine all elliptic curves defined over Q of conductor 11. Firstly, we reduce the problem to one of solving a diophantine equation, namely a certain ThueMahler equation. Then we apply recent sharp inequalities for linear forms in the logarithms of algebraic numbers to bound solutions of that equation. Finally, some straightforward computations yield all solutions of the diophantine equation. Our results are in accordance with the conjecture of Taniyama-Weil for conductor 11. Taniyama and Weil have asked whether all elliptic curves defined over Q of a given conductor N are parametrized by modular functions for the subgroup ro(N) of the modular group. The assertion that this question has a positive answer has become known as the Taniyama-Weil conjecture. While the general question seems shrouded in mystery and quite inaccessible at present, one can at least try to verify the conjecture for small numerical values of N. A considerable amount of work has already been done in this direction (cf. [4], [5], [19] -[24], [29]). However, the first nontrivial case of the conjecture, namely N = 11, has not previously been settled. The aim of this note is to determine all elliptic curves of conductor 11 defined over Q and so to verify the conjecture of Taniyama-Weil for N = 11. It is well known that the problem of finding all elliptic curves defined over Q of a given conductor N can be reduced to finding S-integral points on certain associated curves of genus 1; here S is the set of primes dividing N. For certain values of N, these diophantine equations can easily be solved by congruence techniques. However, this elementary approach does not work for N = 11, and we are forced to solve these equations by using some recent sharp inequalities for linear forms in the logarithms of algebraic numbers. The body of this paper is, thus, given over to solving a diophantine equation by Baker's method. Whilst our computations are of course specific to the particular equation we solve, our methods are quite general. As regards the elliptic curves, we employ the usual notation and terminology. For background and more detailed explanation we refer the reader to the surveys of Swinnerton-Dyer and Birch [31] and of Gelbart [12]; see also Mazur and SwinnertonDyer [18]. 1. An elliptic curve E over a field K has a nonsingular plane cubic model (1) y2 + a1xy + a3y = X3 + a2x2 + a4x + a6 Received July 29, 1979; revised September 19, 1979. 1980 Mathematics Subject Classification. Primary 1OF10, 1OD12, 1OB10, 1OB16, 14K07, 12A30. i 1980 American Mathematical Society 0025-5718/80/0000-01 23/$04.00 991 This content downloaded from 157.55.39.215 on Wed, 31 Aug 2016 04:17:10 UTC All use subject to http://about.jstor.org/terms 992 M. K. AGRAWAL, J. H. COATES, D. C. HUNT AND A. J. VAN DER POORTEN with the ai in K. If the characteristic char K of K is not 2, we can replace 4(2y +a1x + a3) byy and 4x byx to obtain (2) y2 = x3 + b2x2 + 8b4x + 16b6