I tried to calculated the integral points for the following curve by Sage, but after a few hours I didn't receive any answer .


[0,0,0,-1609983754533564186692237854003906250000,0]


How can I calculate integral points for this curve?



Integral points on a special curve
This elliptic curve $E: x^3 - ax$, where
$$
a = 1609983754533564186692237854003906250000
 = 2^4 5^{28} 73 \cdot 97 \cdot 2281 \cdot 390001 \cdot 428801,
$$
has the integral $2$-torsion point $T = (0,0)$, plus
at least $17$ pairs $\pm P = (x,\pm y)$ of nontorsion integral points,
with $(x,y)$ as follows:



```
(-695070141601562500, 33447209632705688476562500000)
(2316289620532226562500, 111461304221023574829101562500000)
(-1577320410156250000, 50354105671882629394531250000)
(1020708122564697265625, 32584910670546829700469970703125) *
(-39433010253906250000, 46579627221641540527343750000)
(40828324902587890625, 48227820848605342864990234375) *
(43474893804931640625, 110347464366896152496337890625) *
(-20492969555664062500, 156163643848510742187500000000)
(-5623634204101562500, 94213113773239135742187500000)
(-154178930664062500, 15755059779357910156250000000)
(-39975982055664062500, 21815100375976562500000000000)
(-3450306250000000000, 74255386668395996093750000000)
(-40124306640625000000, 976627163925170898437500000)
(211550113970947265625, 3021093403700607776641845703125) *
(2520419030914306640625, 126518514595970671176910400390625) *
(318603088420664062500, 5641612672625483917968750000000) *
(19818207721836914062500, 2789945206846541022216796875000000) *

```

These are listed in order of increasing canonical height;
the first six points are in pairs $P,P'$ of equal height,
with each $P' = T - P$ where $T=(0,0)$ as above.
The list is probably complete: the heights ranged from $12.3$ to $30.3$,
and none of the other $15000$ or so point pairs of height at most $100$
is integral.


As suggested in the commentary, proving that there are no other integral
points is feasible but requires rather more effort than one would
normally expend without having quite a good reason. Thanks to the
presence of the $2$-torsion point (and some luck with the $2$-descent,
as implemented in Cremona's **mwrank**), one can compute generators
for the group of rational points on $E$. The rank of this group is $8$,
which is large enough that the proof that there are no more integral points
is not routine.


In case somebody does want to take on this computation, note that the
equation for $E$ is not minimal due to the factors $2^4 5^{28} = 156250^4$
of $a$. For a minimal equation, write $(x,y) = (156250^2 x\_0,156250^3 y\_0)$
and divide through by $156250^4$ to get the minimal form
$E\_0: y\_0^2 = x\_0^3 - a\_0^{\phantom0} x\_0^{\phantom0}$ where
$$a\_0 = a / 156250^4 = 2701104520630058561.$$
Seven of the pairs $(x,\pm y)$ of integral points on $E$
become non-integral on $E\_0$; those are marked with asterisks in the above list.


