  We prove a "height-free" effective isogeny estimate for abelian varieties of
$\mathrm{GL}_2$-type.
  More precisely, let $g\in \mathbb{Z}^+$, $K$ a number field, $S$ a finite set
of places of $K$, and $A,B/K$ $g$-dimensional abelian varieties with good
reduction outside $S$ which are $K$-isogenous and of $\mathrm{GL}_2$-type over
$\overline{\mathbb{Q}}$. We show that there is a $K$-isogeny $A\to B$ of degree
effectively bounded in terms of $g$, $K$, and $S$ only.
  We deduce among other things an effective upper bound on the number of
$S$-integral $K$-points on a Hilbert modular variety.
