How can I determine all integer points of the following equation
$$y^2=x^3+10546$$
I tried Magma with
IntegralPoints(EllipticCurve([0,10546]));
but got the answer that it "could not determine the Mordell-Weil group." What are my options here?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityHow can I determine all integer points of the following equation
$$y^2=x^3+10546$$
I tried Magma with
IntegralPoints(EllipticCurve([0,10546]));
but got the answer that it "could not determine the Mordell-Weil group." What are my options here?
This curve has rank 0 over $\mathbb{Q}$. The 2-descent fails to determine this, because the $2$-torsion subgroup of the Tate-Shafarevich group is non-trivial. Instead, one can compute the $L$-value. One can prove that $L(E,1) = 16 \Omega_{+}$. By Kolyvagin, this implies that the rank is $0$.
Now one just needs to compute the torsion order and since there are no non-trivial torsion points. One gets $E(\mathbb{Q})=\{O\}$ and hence there are no integral points either.