Rational halving quotient bound from root size ($\#(E(\mathbb{Q})/2E(\mathbb{Q}))\le 2^{2m+2}$)

For distinct integral roots $\alpha ,\beta ,\gamma$ with $m=\max (|\alpha |,|\beta |,|\gamma |)$, the finite set in unramified halving fields for a split cubic contains only primes at most $2m$. A quadratic Galois character unramified away from $s$ primes is represented by a signed square-free integer supported on those primes, giving at most $2^{s+1}$ characters. The halving cocycle with rational two-torsion has two independent coordinates in $\mathbb{Z}/2\mathbb{Z}$, so the quotient has at most $2^{2s+2}$ elements. Since $m\ge 1$ and there are at most $m$ primes at most $2m$, this gives $2^{2m+2}$. The bound is deliberately crude but depends only on the root size.