  Using a combination of several powerful modularity theorems and class field
theory we derive a new modularity theorem for semistable elliptic curves over
certain real abelian fields. We deduce that if $K$ is a real abelian field of
conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every
semistable elliptic curve $E$ over $K$ is modular.
  Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a
putative non-trivial primitive solution of the generalized Fermat
$x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over
$\mathbb{Q}(\zeta_p)^+$, and study its mod $\ell$ representation with the help
of level lowering and our modularity result. We deduce the non-existence of
non-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne
7$.
