 Torsion alternative for opposite quartic-twist coefficients

Suppose $15\nmid n$ and both $n,-n$ are nonsquares. Each elliptic curve has just one nonidentity rational 2-torsion point, $(0,0)$. The duplication formula shows that halving it requires the corresponding coefficient $\pm n$ to be a square, so neither has rational order-four points. If $3\nmid n$, both good reductions at three have order four, forcing torsion point of an elliptic curve to be exactly order two. Otherwise $5\nmid n$; the reduction orders for coefficients $1,2,3,4$ modulo five are $4,2,10,8$. Among opposite coefficients at least one has order two or four, giving the same conclusion for at least one curve by reduction of torsion points on an elliptic curve.