LMFDB - Elliptic curve with Cremona label 266910v2 (LMFDB label 266910.v2)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+xy+y=x^3+128386x-14268688$ y^2+xy+y=x^3+128386x-14268688	(homogenize, simplify)
$y^2z+xyz+yz^2=x^3+128386xz^2-14268688z^3$ y^2z+xyz+yz^2=x^3+128386xz^2-14268688z^3	(dehomogenize, simplify)
$y^2=x^3+166388877x-666219062322$ y^2=x^3+166388877x-666219062322	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 1, 128386, -14268688])

gp: E = ellinit([1, 0, 1, 128386, -14268688])

magma: E := EllipticCurve([1, 0, 1, 128386, -14268688]);

oscar: E = elliptic_curve([1, 0, 1, 128386, -14268688])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Invariants

Conductor:	$N$	=	$266910$	=	$2 \cdot 3 \cdot 5 \cdot 7 \cdot 31 \cdot 41$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$-223522342029504000$	=	$-1 \cdot 2^{9} \cdot 3^{5} \cdot 5^{3} \cdot 7 \cdot 31^{3} \cdot 41^{3}$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$\frac{234035413953867370151}{223522342029504000}$	=	$2^{-9} \cdot 3^{-5} \cdot 5^{-3} \cdot 7^{-1} \cdot 17^{3} \cdot 23^{3} \cdot 31^{-3} \cdot 41^{-3} \cdot 15761^{3}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$ (no potential complex multiplication)			sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$2.0161412160281481437846393181$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$2.0161412160281481437846393181$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$0.9228736675494511$
Szpiro ratio:	$\sigma_{m}$	≈	$3.7537618940206223$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Mordell-Weil rank:	$r$	=	$0$	comment: Rank sage: E.rank() gp: [lower,upper] = ellrank(E) magma: Rank(E);
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$0.17180260480970568954051099144$	comment: Real Period sage: E.period_lattice().omega() gp: if(E.disc>0,2,1)E.omega[1] magma: (Discriminant(E) gt 0 select 2 else 1) RealPeriod(E);
Tamagawa product:	$\prod_{p}c_p$	=	$15$ = $1\cdot5\cdot1\cdot1\cdot3\cdot1$	comment: Tamagawa numbers sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] \| i<-[1..#gr[4][,1]]] magma: TamagawaNumbers(E); oscar: tamagawa_numbers(E)
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L(E,1)$	≈	$2.5770390721455853431076648716$	comment: Special L-value r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial() gp: [r,L1r] = ellanalyticrank(E); L1r/r! magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
Analytic order of Ш:	Ш ${}_{\mathrm{an}}$	=	$1$ (exact)	comment: Order of Sha sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E);

$\displaystyle 2.577039072 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.171803 \cdot 1.000000 \cdot 15}{1^2} \approx 2.577039072$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

Modular invariants

Modular form 266910.2.a.v

q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} - q^{14} - q^{15} + q^{16} - q^{18} + 8 q^{19} + O(q^{20})

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

For more coefficients, see the Downloads section to the right.

Modular degree:	3538080	comment: Modular degree sage: E.modular_degree() gp: ellmoddegree(E) magma: ModularDegree(E);
$\Gamma_0(N)$ -optimal:	no
Manin constant:	1	comment: Manin constant magma: ManinConstant(E);

Local data at primes of bad reduction

This elliptic curve is semistable. There are 6 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{9}$	nonsplit multiplicative	1	1	9	9
$3$	$5$	$I_{5}$	split multiplicative	-1	1	5	5
$5$	$1$	$I_{3}$	nonsplit multiplicative	1	1	3	3
$7$	$1$	$I_{1}$	split multiplicative	-1	1	1	1
$31$	$3$	$I_{3}$	split multiplicative	-1	1	3	3
$41$	$1$	$I_{3}$	nonsplit multiplicative	1	1	3	3

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

Galois representations

The $\ell$ -adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod- $\ell$ image	$\ell$ -adic image
$3$	3B.1.2	3.8.0.2

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[533821, 6, 533823, 19], [4, 3, 9, 7], [1067635, 6, 1067634, 7], [610081, 6, 762603, 19], [427057, 6, 213531, 19], [26041, 6, 78123, 19], [998761, 6, 861003, 19], [266911, 6, 0, 1], [1, 6, 0, 1], [489338, 578307, 1, 934186], [3, 4, 8, 11], [1, 0, 6, 1]]

GL(2,Integers(1067640)).subgroup(gens)

Gens := [[533821, 6, 533823, 19], [4, 3, 9, 7], [1067635, 6, 1067634, 7], [610081, 6, 762603, 19], [427057, 6, 213531, 19], [26041, 6, 78123, 19], [998761, 6, 861003, 19], [266911, 6, 0, 1], [1, 6, 0, 1], [489338, 578307, 1, 934186], [3, 4, 8, 11], [1, 0, 6, 1]];

sub<GL(2,Integers(1067640))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $1067640 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 31 \cdot 41$ , index $16$ , genus $0$ , and generators

$\left(\begin{array}{rr} 533821 & 6 \\ 533823 & 19 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1067635 & 6 \\ 1067634 & 7 \end{array}\right),\left(\begin{array}{rr} 610081 & 6 \\ 762603 & 19 \end{array}\right),\left(\begin{array}{rr} 427057 & 6 \\ 213531 & 19 \end{array}\right),\left(\begin{array}{rr} 26041 & 6 \\ 78123 & 19 \end{array}\right),\left(\begin{array}{rr} 998761 & 6 \\ 861003 & 19 \end{array}\right),\left(\begin{array}{rr} 266911 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 489338 & 578307 \\ 1 & 934186 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$ .

The torsion field $K:=\Q(E[1067640])$ is a degree- $10968608786507366400000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1067640\Z)$ .

The table below list all primes $\ell$ for which the Serre invariants associated to the mod- $\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	nonsplit multiplicative	$4$	$133455 = 3 \cdot 5 \cdot 7 \cdot 31 \cdot 41$
$3$	split multiplicative	$4$	$7$
$5$	nonsplit multiplicative	$6$	$17794 = 2 \cdot 7 \cdot 31 \cdot 41$
$7$	split multiplicative	$8$	$38130 = 2 \cdot 3 \cdot 5 \cdot 31 \cdot 41$
$31$	split multiplicative	$32$	$8610 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 41$
$41$	nonsplit multiplicative	$42$	$6510 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 31$

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 266910v consists of 2 curves linked by isogenies of degree 3.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	$\Q(\sqrt{-3})$	$\Z/3\Z$	not in database
$3$	3.1.1067640.1	$\Z/2\Z$	not in database
$3$	3.1.11907.1	$\Z/3\Z$	not in database
$6$	6.0.1216954973271744000.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$6$	6.0.425329947.3	$\Z/3\Z \oplus \Z/3\Z$	not in database
$6$	6.0.3419565508800.2	$\Z/6\Z$	not in database
$9$	9.1.4658466548547515629058112000.1	$\Z/6\Z$	not in database
$12$	deg 12	$\Z/4\Z$	not in database
$12$	deg 12	$\Z/2\Z \oplus \Z/6\Z$	not in database
$18$	18.0.15450049497024747368006083228126647233457361330605011042877685546875.1	$\Z/9\Z$	not in database
$18$	18.0.65103931751808608367745669092412956292404819013632000000.1	$\Z/3\Z \oplus \Z/6\Z$	not in database
$18$	18.0.59885527985568927080348437848457311286109110372280931466346496000000000.1	$\Z/2\Z \oplus \Z/6\Z$	not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$ -adic regulators

All $p$ -adic regulators are identically $1$ since the rank is $0$ .

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

Properties

Related objects

Downloads

Learn more

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

Mordell-Weil group structure

Invariants

BSD invariants

BSD formula

Modular invariants

Local data at primes of bad reduction

Galois representations

Isogenies

Twists

Growth of torsion in number fields

Iwasawa invariants

$p$ -adic regulators

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	266910v2

Conductor	$266910$
Discriminant	$-2.235\times 10^{17}$
j-invariant	$\frac{234035413953867370151}{223522342029504000}$
CM	no
Rank	$0$
Torsion structure	trivial

Properties

Related objects

Downloads

Learn more

Mordell-Weil group structure

Invariants

Local data at primes of bad reduction

Growth of torsion in number fields

ppp-adic regulators

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

$p$ -adic regulators