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?
$\begingroup$
$\endgroup$
2
-
1$\begingroup$ 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 SleziakCommented Jan 6, 2020 at 13:40
-
1$\begingroup$ 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 $y^2=x^3+k.$) $\endgroup$– castorCommented Jan 6, 2020 at 18:32
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
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.
answered Jan 6, 2020 at 13:53
-
$\begingroup$ Nice! Can you please point to the origin of $L(E,1)$. $\endgroup$– yarchikCommented Jan 6, 2020 at 20:59
-
1$\begingroup$ 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. $\endgroup$ Commented Jan 6, 2020 at 21:27