A database of genus-2 curves over the rational numbers

Booker, Andrew R.; Sijsling, Jeroen; Sutherland, Andrew V.; Voight, John; Yasaki, Dan

doi:10.1112/s146115701600019x

Cited by 49 publications

(83 citation statements)

References 32 publications

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“…Six curves in the first dictionary have |J 10 | < 10000. It would be interesting to see exactly how the database in [3] and [5] intersect.…”

Section: Constructing the Databasesmentioning

confidence: 99%

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Rational points in the moduli space of genus two

Beshaj¹,

Hidalgo²,

Kruk³

et al. 2018

Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

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We build a database of genus 2 curves defined over Q which contains all curves with minimal absolute height h ≤ 5, all curves with moduli height h ≤ 20, and all curves with extra automorphisms in standard form y 2 = f (x 2 ) defined over Q with height h ≤ 101. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space M 2 for which the field of moduli is a field of definition is discussed and some open problems are presented.

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“…Six curves in the first dictionary have |J 10 | < 10000. It would be interesting to see exactly how the database in [3] and [5] intersect.…”

Section: Constructing the Databasesmentioning

confidence: 99%

“…It improves and expands a previous Maple genus 2 computational algebra package as in [21]. There is another database of genus 2 curves in [5] which collects all genus 2 curves with discriminants ≤ 1000. Some remarks on how the two databases overlap can be found in the last section.…”

Section: Introductionmentioning

confidence: 99%

Rational points in the moduli space of genus two

Beshaj¹,

Hidalgo²,

Kruk³

et al. 2018

Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

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“…The most current version of Cremona's tables, and similar tables of elliptic curves over various number fields, can be found in the L-functions and modular forms database (LMFDB) [26]. Motivated by the utility of Cremona's tables, the LMFDB now includes a table of genus 2 curves over Q whose construction is described in [1]. The goal of this article is to describe the first steps toward the construction of a similar table of genus 3 curves over Q.…”

Section: Introductionmentioning

confidence: 99%

“…The methods used in [1] extend fairly easily to genus 3 hyperelliptic curves and have been used to construct a list of genus 3 hyperelliptic curves over Q of small discriminant, and to compute their conductors, Euler factors at bad primes, endomorphism rings, and Sato-Tate groups. We plan to make this data available in the LMFDB later this year (2018); a preliminary list of these curves can be found at the author's website.…”

Section: Introductionmentioning

confidence: 99%

A database of nonhyperelliptic genus-3 curves over ℚ

Sutherland¹

2019

Open Book Series

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We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant. of this database: an enumeration of all smooth plane quartic curves with coefficients of absolute value at most B c := 9, with the aim of obtaining a set of unique Q-isomorphism class representatives for all such curves that have absolute discriminant at most B ∆ := 10 7 .Even after accounting for obvious symmetries, this involves more than 10 17.5 possible curve equations and requires a massively distributed computation to complete in a reasonable amount of time. Efficiently computing the discriminants of these equations is a non-trivial task, much more so than in the hyperelliptic case, and much of this article is devoted to an explanation of how this was done. Many of the techniques that we use can be generalized to other enumeration problems and may be of independent interest, both from an algorithmic perspective, and as an example of how cloud computing can be effectively applied to a research problem in number theory. A list of the curves that were found (more than 80 thousand) is available on the author's website [33].Remark 1.1. The informed reader will know that not every genus 3 curve over Q falls into the category of smooth plane quartics f (x, y, z) = 0 or curves with a hyperelliptic model y 2 + h(x) y = f (x). The other possibility is a degree-2 cover of a pointless conic; see [18] for a discussion of such curves and algorithms to efficiently compute their L-functions. We plan to conduct a separate search for curves of this form that will also become part of the genus 3 database in the LMFDB.1.1. Acknowledgments. The author is grateful to Nils Bruin, Armand Brumer, John Cremona, Tim Dokchitser, Jeroen Sijsling, Michael Stoll, and John Voight for their insight and helpful comments, and to the anonymous referees for their careful reading and suggestions for improvement. THE DISCRIMINANT OF A SMOOTH PLANE CURVELet C[x] d denote the space of ternary forms of degree d ≥ 1, as homogeneous polynomials in the variables x := (x 0 , x 1 , x 2 ). It is a C-vector space of dimension n d := d+2 2 equipped with a standard monomial basis

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“…The second application is computing zeros and special values of L-functions to high precision; this played an important role in the recent addition of genus two curves to the L-functions and modular forms database (LMFDB) [6], as described in [1] (as noted above, this application also requires the Euler factors at primes of bad reduction).…”

Section: Introductionmentioning

confidence: 99%

Computing -series of geometrically hyperelliptic curves of genus three

Harvey¹,

Massierer²,

Sutherland³

2016

LMS J. Comput. Math.

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Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N . The algorithm relies on an adaptation of the 'accumulating remainder tree' to matrices with entries in a quadratic field. We report on an implementation and compare its performance to previous algorithms for the ordinary hyperelliptic case.

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A database of genus-2 curves over the rational numbers

Abstract: We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function.

Cited by 49 publications

References 32 publications

Rational points in the moduli space of genus two

Rational points in the moduli space of genus two

A database of nonhyperelliptic genus-3 curves over ℚ

Computing -series of geometrically hyperelliptic curves of genus three

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