  Using methods from analytic number theory, for $m > 5$ and for $m = 4$, we
obtain asymptotics with power-saving error terms for counts of elliptic curves
with a cyclic $m$-isogeny up to quadratic twist over the rational numbers. For
$m > 5$, we then apply a Tauberian theorem to achieve asymptotics with power
saving error for counts of elliptic curves with a cyclic $m$-isogeny up to
isomorphism over the rational numbers.
