Rank and generators of elliptic curve

Question

I need to calculate the rank and the generators of the elliptic curve

            [0,1,0,-15662264585,746984342506759]

that is, $$ y^2 = x^3 + x^2 -15662264585 x + 746984342506759. $$

Answer 1

For the record, mwrank eventually confirmed that this curve $$ E : y^2 = x^3 + x^2 - 15662264585 x + 746984342506759 $$ has rank $8$, as suggested by the calculation of integral points ($28$ pairs $(x,\pm y)$ with $x < 10^8$, but all in the same rank-8 group) and corroborated by the global root number of $+1$ (which under the parity conjecture implies that the rank is even, so if the rank exceeds $8$ there must be at least two further independent generators that do not contribute to the list of small integral points).

Shortly after this question was posed, I went to germain.math.harvard.edu and set John Cremona's program mwrank on this curve $E$. Some hours later, mwrank had found six independent points while

Looking for Type 2 quartics:
Trying positive a from 1 up to 70218 (square a first...)

but then seemed to go into hibernation after announcing it was

Trying positive a from 1 up to 70218 (...then non-square a)

Still, there was nothing else happening on germain, so I let the process continue, and then forgot about it for some time. Recently I checked again and found that the program had completed the calculation three weeks(!) later, finishing its report with

Generator 1 is [78271:-777724:1]; height 9.80876524759791
Generator 2 is [78077:294276:1]; height 10.1492862271008
Generator 3 is [10092319809:21622801863508:2197]; height 19.7776896372427
Generator 4 is [4284948894:19048354703:54872]; height 18.1180101559044
Generator 5 is [6394749009:2154538281608:35937]; height 18.3383217073227
Generator 6 is [8109376287:42485008616:103823]; height 17.8505437330629
Generator 7 is [5638548716266233736152:-615800483663476330089571:98759759636551168]; height 37.1133372519458
Generator 8 is [73421481:29109699172:343]; height 15.0939038726812

Regulator = 6202462.51465316

The rank and full Mordell-Weil basis have been determined unconditionally.

The full output is here.

There must be faster ways of determining the rank and Mordell-Weil group of an elliptic curve with coefficients, rank, and generators of this size, but the fact that mwrank was able to do it in three weeks might still be an interesting data point about this kind of computation.

The question at the end of my comment still stands:

This still begs the question of why this curve is of interest (yes, rank $\geq 8$ is unusual, but we've known for some time how to generate infinitely many such curves) and how the OP found it.

Answer 2

As another data point, Magma was able to find 8 generators in about 15 seconds under the assumption of GRH.

SetClassGroupBounds("GRH");
E := EllipticCurve([0,1,0,-15662264585,746984342506759]);
Generators(E);

Output:

[ (29182241521/369664 : -357905489312311/224755712 : 1), (100099 : -13497788 : 1), (72219471 : 613734879724 : 1), (162669513/1369 : 1202545309672/50653 : 1), (52676723 : 382319900340 : 1), (-15653710/121 : -32820199443/1331 : 1), (20638562/169 : -56252791245/2197 : 1), (56710535981/722500 : -677498563801629/614125000 : 1) ]
true true

Calculations are restricted to 120 seconds.
Input is limited to 50000 bytes.
Running Magma V2.24-1.
Seed: 2015558728; Total time: 15.919 seconds; Total memory usage: 149.22MB.

Answer 3

Further update: gp now has a command ellrank that finds the rank quickly:

allocatemem(2^26)
E = ellinit([0,1,0,-15662264585,746984342506759]);
ellrank(E)

takes under a second on my laptop to reply

[8, 8, 0, [[78077, 294276], [116479, 22427332], [-1085129/9, 802159084/27], [-3855/49, 9382266704/343], [11662649/121, 15339215952/1331], [42354193/289, 196480125128/4913], [2521274/25, 1737395457/125], [112761813/1444, 19048354703/54872]]]

The first two numbers $r_1,r_2$ in the output are lower and upper bounds on the rank; here $r_1 = r_2 = 8$ so this curve has rank 8. The third number encodes information about the Tate-Shafarevic group Sha; here it says that Sha[2] is trivial. The last output is an LLL-reduced $\bf Z$-basis $B$ for a rank-$r_1$ subgroup of the Mordell-Weil group E(Q).

This function was added by Bill Allombert in early 2021 according to https://pari.math.u-bordeaux.fr/archives/pari-dev-2102/msg00003.html, but I only learned of it last month from John Cremona. The online documentation at https://pari.math.u-bordeaux.fr/dochtml/html/Elliptic_curves.html says that the 2-Selmer rank is computed unconditionally, which would mean that the bounds $r_1,r_2$ do not depend on GRH or on any other unproved hypothesis.

$\langle B \rangle$ is not necessarily saturated, and indeed in this case the function ellsaturation finds a $\bf Z$-basis $B'$, again LLL-reduced, for a larger group of rational points that contains $\langle B \rangle$ with index 5:

[[116479, 22427332], [17023, -22029500], [-139583, 14616304], [52797, 8199964],
[-143366, 6761923], [31635, -16827628], [66259, 341548], [66305, 49728]]

Unlike mwrank, this program will not certify that the result is fully saturated: ellsaturation requires an upper bound on the saturation primes. Still it takes about a second to certify that the index is not divisible by any prime less than $2^{12}$; so it's morally certain --- and likely provable --- that $\langle B' \rangle$ is the full Mordell-Weil group.