Elliptic curves of conductor 11

Agrawal, Megha; Coates, John; Hunt, Douglas; Poorten, Alfred J. van der

doi:10.1090/s0025-5718-1980-0572871-5

Cited by 14 publications

(3 citation statements)

References 21 publications

(17 reference statements)

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“…To give the reader an idea of our running times, we now discuss some equations appearing in the literature. We solved the equation of Tzanakis-de Weger [TdW91], and we deter-mined all solutions of the equation of Agraval-Coates-Hunt-van der Poorten [ACHvdP80].…”

Section: Applicationsmentioning

confidence: 99%

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

Känel¹,

Matschke²

2016

Preprint

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In the first part we construct algorithms (over Q) which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. The first type of our algorithms uses modular symbols, and the second type combines explicit height bounds with efficient sieves. In particular we construct a refined sieve for S-unit equations which combines Diophantine approximation techniques of de Weger with new geometric ideas. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we used the resulting data to motivate various conjectures and questions, including Baker's explicit abc-conjecture and a new conjecture on S-integral points of any hyperbolic genus one curve over Q.In the second part we establish new results for certain old Diophantine problems (e.g. the difference of squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and Vinogradov-Sprindžuk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases. We also conduct some effort to work out optimized height bounds for S-unit and Mordell equations which are used in our algorithms of the first part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Paršin, Szpiro) combined with the Shimura-Taniyama conjecture, and they all do not use lower bounds for linear forms in (elliptic) logarithms.In the third part we solve the problem of constructing an efficient sieve for the S-integral points of bounded height on any elliptic curve E over Q with given Mordell-Weil basis of E(Q). Here we combine a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier) with several conceptually new ideas. The resulting "elliptic logarithm sieve" is crucial for some of our algorithms of the first part. Moreover, it considerably extends the class of elliptic Diophantine equations which can be solved in practice: To demonstrate this we solved many notoriously difficult equations by combining our sieve with known height bounds based on the theory of logarithmic forms.

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Section: Applicationsmentioning

confidence: 99%

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

Känel¹,

Matschke²

2016

Preprint

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“…We have computed a subset of this is heuristically the full set, but is not proved to be complete by our method at present. 1 In Sections 1.1 and 1.2 we give a summary of our data and discuss some statistics of the data. We compare our data to Cremona's database in Section 1.3.…”

Section: Introductionmentioning

confidence: 99%

“…The set M ({2, 3}) was computed by Coghlan [9] and Stephens [37], and Coghlan's data was republished as Table 4 in [6]. Agrawal, Coates, Hunt and van der Poorten [1] computed M ({11}) via a reduction to Thue-Mahler equations. Cremona and Lingham [13] computed M ({2, p}) for p ≤ 23 via a reduction to the computation of S-integral points on Mordell curves.…”

Section: Introductionmentioning

confidence: 99%

Elliptic curves with good reduction outside of the first six primes

Best¹,

Matschke²

2020

Preprint

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We present a database of rational elliptic curves, up to Q-isomorphism, with good reduction outside {2, 3, 5, 7, 11, 13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor.

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The Last Period

Narkiewicz

2012

Springer Monographs in Mathematics

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Elliptic curves of conductor 11

Cited by 14 publications

References 21 publications

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

Elliptic curves with good reduction outside of the first six primes

The Last Period

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