= Torsion alternative for opposite quartic-twist coefficients
{title2=$E_\pm:y^2=x^3\pm nx$}
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>.
Back to article page
