Properties

Label 2730.bd5
Conductor 2730
Discriminant 21894701746029840
j-invariant \( \frac{443915739051786565201}{21894701746029840} \)
CM no
Rank 0
Torsion Structure \(\Z/{4}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -158925, -23336703]); // or
 
magma: E := EllipticCurve("2730bd3");
 
sage: E = EllipticCurve([1, 0, 0, -158925, -23336703]) # or
 
sage: E = EllipticCurve("2730bd3")
 
gp: E = ellinit([1, 0, 0, -158925, -23336703]) \\ or
 
gp: E = ellinit("2730bd3")
 

\( y^2 + x y = x^{3} - 158925 x - 23336703 \)

Mordell-Weil group structure

\(\Z/{4}\Z\)

Torsion generators

magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 

\( \left(-228, 1143\right) \)

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 

\( \left(-228, 1143\right) \), \( \left(458, -229\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 2730 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(21894701746029840 \)  =  \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{12} \cdot 13^{3} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{443915739051786565201}{21894701746029840} \)  =  \(2^{-4} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-12} \cdot 11^{3} \cdot 13^{-3} \cdot 37^{3} \cdot 18743^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(0\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.239843976092\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 288 \)  = \( 2^{2}\cdot2\cdot1\cdot( 2^{2} \cdot 3 )\cdot3 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(4\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 2730.2.a.bd

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + q^{12} + q^{13} + q^{14} + q^{15} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 27648
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L(E,1) \) ≈ \( 4.31719156965 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(3\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(5\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(7\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(13\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right)$ and has index 12.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B.1.2

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7 13
Reduction type split split split split split
$\lambda$-invariant(s) 3 11 1 1 1
$\mu$-invariant(s) 0 1 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 2730.bd consists of 8 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{65}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-3}) \) \(\Z/12\Z\) Not in database
3 3.1.24300.4 \(\Z/12\Z\) Not in database
4 \(\Q(\sqrt{-3}, \sqrt{65})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
4.0.1834560.5 \(\Z/8\Z\) Not in database
6 6.0.1771470000.2 \(\Z/3\Z \times \Z/12\Z\) Not in database
6.2.6486532650000.18 \(\Z/2\Z \times \Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.