  Clemm and Trebat-Leder (2014) proved that the number of quadratic number
fields with absolute discriminant bounded by $x$ over which there exist
elliptic curves with good reduction everywhere and rational $j$-invariant is
$\gg x\log^{-1/2}(x)$. In this paper, we assume the $abc$-conjecture to show
the sharp asymptotic $\sim cx\log^{-1/2}(x)$ for this number, obtaining
formulae for $c$ in both the real and imaginary cases. Our method has three
ingredients:
  (1) We make progress towards a conjecture of Granville: Given a fixed
elliptic curve $E/\mathbb{Q}$ with short Weierstrass equation $y^2 = f(x)$ for
reducible $f \in \mathbb{Z}[x]$, we show that the number of integers $d$, $|d|
\leq D$, for which the quadratic twist $dy^2 = f(x)$ has an integral
non-$2$-torsion point is at most $D^{2/3+o(1)}$, assuming the $abc$-conjecture.
  (2) We apply the Selberg--Delange method to obtain a Tauberian theorem which
allows us to count integers satisfying certain congruences while also being
divisible only by certain primes.
  (3) We show that for a polynomially sparse subset of the natural numbers, the
number of pairs of elements with least common multiple at most $x$ is
$O(x^{1-\epsilon})$ for some $\epsilon > 0$. We also exhibit a matching lower
bound.
  If instead of the $abc$-conjecture we assume a particular tail bound, we can
prove all the aforementioned results and that the coefficient $c$ above is
greater in the real quadratic case than in the imaginary quadratic case, in
agreement with an experimentally observed bias.
