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

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

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

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    $\begingroup$ Why this curve? I'm guessing you meant [0, 1, 0, -15662264585, 746984342506759] (missing the third coefficient 0), because this curve has 28 pairs of integral points with x < 10^8, which generate a group of rank 8, and that kind of thing almost never happens at random . . . $\endgroup$ Apr 5, 2017 at 19:58
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    $\begingroup$ I see that this question has been put on hold on the grounds "unclear what you're asking". It is by now clear that the intended elliptic curve was $y^2 = x^3 + x^2 -15662264585 x + 746984342506759$ (using the standard notation $[a_1,a_2,a_3,a_4,a_6]$ [sic] for the curve with extended Weierstrass form $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$). So I'm editing the question and voting to reopen. 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. $\endgroup$ Apr 5, 2017 at 20:43

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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.

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  • $\begingroup$ Sorry, pet peeve: en.wikipedia.org/wiki/Begging_the_question $\endgroup$
    – Lucia
    Jul 10, 2017 at 20:22
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    $\begingroup$ Well the phrase "petitio principii" hardly made sense in Latin, and its translation as "beg the question" makes even less sense in English for a form of circular reasoning. So it's natural for the phrase to be used as if it were "beg for the question". By now even Merriam-Webster recognizes "to elicit a question logically as a reaction or response" as the primary sense of the phrase, with secondary sense "to pass over or ignore a question by assuming it to be established or settled". merriam-webster.com/dictionary/beg%20the%20question $\endgroup$ Dec 4, 2018 at 22:07
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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.
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