  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 $\mathbb{Q}$ 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
$\mbox{PGL}_2(\mathbb{F}_l)$. By constructing models of two modular curves,
$X_{\text{s}}(5)$ and $X_{S_4}(13)$, 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.
