= Rational halving quotient bound from root size
{title2=$\#(E(\mathbb Q)/2E(\mathbb Q))\leq 2^{2m+2}$}
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 $m\geq1$ 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.
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