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Number of elliptic curves over $\mathbb{Q}$ with good redution outside $S$
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Numbers
$S$
count
INPUT{numbers.yaml} (not shown in preview)
Definition
Let $S$ be a finite set of primes. An elliptic curve $E/\mathbb{Q}$ is said to have good reduction outside $S$ if its minimal discriminant (equivalently, its conductor) is only divisible by primes in $S$. This table lists the numbers of elliptic curves $E/\mathbb{Q}$ with good reduction outside $S$ counted in two different ways: up to $\mathbb{Q}$-isomorphism, and up to $\overline{\mathbb{Q}}$-isomorphism.
Parameters
$S$
—   set of primes
Comments
(1)
Elliptic curves over $\mathbb{Q}$ with good reduction outside of the first $n$ have been computed by several authors: $n=0$ by Tate (see [6]), $n=1$ by Ogg [6], $n=2$ by Coghlan [3] and Stephens [7], $n=3,4,5$ by von Känel and Matschke [4] and Bennett, Gherga and Rechnitzer [1], $n=6$ heuristically by Best and Matschke [2], $n=7,8,9^*$ by Matschke [9] ($n=9$ assumes GRH).
References
[1]
Michael A. Bennett, Adela Gherga, and Andrew Rechnitzer, "Computing elliptic curves over Q". Math. Comp., 88(317):1341–1390, 2019.
[2]
A. Best, B. Matschke, "Elliptic curves with good reduction outside of the first six primes". 2020 (github)
[3]
Francis Coghlan, "Elliptic Curves with Conductor $2^m 3^n$". Ph.D. thesis, Manchester, England, 1967.
[4]
R. von Känel, B. Matschke, "Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture". 2016 (github)
[5]
Angelos Koutsianas, "Computing all elliptic curves over an arbitrary number field with prescribed primes of bad reduction". Exp. Math., 28(1):1–15, 2019. (arXiv) (github)
[6]
Andrew P. Ogg, "Abelian curves of 2-power conductor". Math. Proc. Camb. Philos. Soc., 62(2):143–148, 1966.
[7]
Nelson M. Stephens, "The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank". Ph.D. Thesis, Manchester, 1965.
Links
Data properties
Entries are of type: real number
Table is complete: no
Sources of data: [9]
Reliability: Assumes GRH if $23\in S$. Needs to be recomputed to be provably correct (see readme.md in [9]).