Tetrahedral elliptic curves and the local-global principle for isogenies

We study the failure of a local-global principle for the existence of l-isogenies for elliptic curves over number fields K. Sutherland has shown that over Q there is just one failure, which occurs for l = 7 and a unique j-invariant, and has given a classification of such failures when K does not contain the quadratic subfield of the l-th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new "exceptional" source of such failures arising from the exceptional subgroups of PGL(2)(F-l). By constructing models of two modular curves, X-s(5) and X-S4(13), we find two new families of elliptic curves for which the principle fails, and we show that, for quadratic fields, there can be no other exceptional failures.