# Integer points of one Mordell equation

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?

• I have edited the tags a bit - it is recommended to use at least one top level tag on MathOverflow. I wasn't entirely sure whether or not to include the tag (magma) - feel free to remove it if you think it does not fit. Jan 6 '20 at 13:40
• There are some nice papers containing data related to the question: Gebel, Pethő and Zimmer: On Mordell’s equation, Compos. Math. 110, No. 3, 335-367 (1998) and Bennett and Ghadermarzi: Mordell’s equation: a classical approach, LMS J. Comput. Math. 18, 633-646 (2015) (here the technique is applied in case of $|k|<10^7$ to the equation $y^2=x^3+k.$) Jan 6 '20 at 18:32

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.
• Nice! Can you please point to the origin of $L(E,1)$. Jan 6 '20 at 20:59
• sage, magma, pari,... they can all compute $L$-function with sufficient precision to convince you that $L(E,1)\neq 0$. For the precise proven value of 16 I used my own code. Jan 6 '20 at 21:27