LMFDB - Elliptic curve search results

Refine search

Base field	Conductor norm	Rank*	Torsion order	CM discriminant

Base field degree/signature	Bad $p$	$\Q$-curves	Torsion structure	CM

Analytic order of Ш*	Cyclic isogeny degree	Isogeny class size	Reduction	Galois image

j-invariant	Regulator*	Curves per isogeny class	Isogeny class degree	Nonmax$\ \ell$

		Sort order	Select

Results (displaying both matches)

Download displayed columns for results to

Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
31.2-a2	31.2-a	$6$	$8$	$\Q(\zeta_{15})^+$	$4$	$[4, 0]$	31.2	$ 31 $	$ 31^{4} $	$4.60400$	$(-a^3+5a-2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$ 2 $	$1$	$226.2402485$	0.843147624	$ -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} $	$ \bigl[a + 1$ , $ a^{3} - 3 a$ , $ a^{2} - 1$ , $ 270 a^{3} - 70 a^{2} - 655 a - 231$ , $ -2943 a^{3} + 1542 a^{2} + 6186 a + 1507\bigr] $	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(270a^{3}-70a^{2}-655a-231\right){x}-2943a^{3}+1542a^{2}+6186a+1507$
31.2-b1	31.2-b	$6$	$8$	$\Q(\zeta_{15})^+$	$4$	$[4, 0]$	31.2	$ 31 $	$ 31^{4} $	$4.60400$	$(-a^3+5a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$16$	$ 2 $	$1$	$2.959428063$	0.705864781	$ -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} $	$ \bigl[a^{3} - 2 a + 1$ , $ a^{3} - a^{2} - 2 a + 2$ , $ a^{2} - 1$ , $ 416 a^{3} + 76 a^{2} - 1585 a - 348$ , $ 6396 a^{3} + 1587 a^{2} - 23865 a - 5050\bigr] $	${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(416a^{3}+76a^{2}-1585a-348\right){x}+6396a^{3}+1587a^{2}-23865a-5050$

Download displayed columns for results to

*The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Learn more

Refine search

Results (displaying both matches)

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
31.2-a2	31.2-a	$6$	$8$	\(\Q(\zeta_{15})^+\)	$4$	$[4, 0]$	31.2	\( 31 \)	\( 31^{4} \)	$4.60400$	$(-a^3+5a-2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 2 \)	$1$	$226.2402485$	0.843147624	\( -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} \)	\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 270 a^{3} - 70 a^{2} - 655 a - 231\) , \( -2943 a^{3} + 1542 a^{2} + 6186 a + 1507\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(270a^{3}-70a^{2}-655a-231\right){x}-2943a^{3}+1542a^{2}+6186a+1507$
31.2-b1	31.2-b	$6$	$8$	\(\Q(\zeta_{15})^+\)	$4$	$[4, 0]$	31.2	\( 31 \)	\( 31^{4} \)	$4.60400$	$(-a^3+5a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$16$	\( 2 \)	$1$	$2.959428063$	0.705864781	\( -\frac{29296819153015039230435114609}{923521} a^{3} - \frac{9909977575470350951831216400}{923521} a^{2} + \frac{103925138003905943940922438084}{923521} a + \frac{21891704604902707693397138361}{923521} \)	\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - 1\) , \( 416 a^{3} + 76 a^{2} - 1585 a - 348\) , \( 6396 a^{3} + 1587 a^{2} - 23865 a - 5050\bigr] \)	${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(416a^{3}+76a^{2}-1585a-348\right){x}+6396a^{3}+1587a^{2}-23865a-5050$