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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?

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    $\begingroup$ 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. $\endgroup$ – Martin Sleziak Jan 6 at 13:40
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    $\begingroup$ 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 $y^2=x^3+k.$) $\endgroup$ – castor Jan 6 at 18:32
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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.

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  • $\begingroup$ Nice! Can you please point to the origin of $L(E,1)$. $\endgroup$ – yarchik Jan 6 at 20:59
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    $\begingroup$ 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. $\endgroup$ – Chris Wuthrich Jan 6 at 21:27

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