On the number of n-isogenies of elliptic curves over number fields

Abstract

Abstract. We find the number of elliptic curves with a cyclic isogeny of degree n over various number fields by studying the modular curves X 0 (n). We show that for n = 14, 15, 20, 21, 49 there exist infinitely many quartic fields K such that #Y 0 (n)(Q) = #Y 0 (n)(K) < ∞. In the case n = 27 we prove that there are infinitely many sextic fields such that #Y 0 (n)(Q) = #Y 0 (n)(K) < ∞.