Properties

Label 14490.by4
Conductor $14490$
Discriminant $-1.997\times 10^{22}$
j-invariant \( \frac{47342661265381757089751}{27397579603968000000} \)
CM no
Rank $1$
Torsion structure \(\Z/{6}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 6782953, -20348881])
 
gp: E = ellinit([1, -1, 1, 6782953, -20348881])
 
magma: E := EllipticCurve([1, -1, 1, 6782953, -20348881]);
 

\(y^2+xy+y=x^3-x^2+6782953x-20348881\)  Toggle raw display

Mordell-Weil group structure

\(\Z\times \Z/{6}\Z\)

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \(\left(187, 35326\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $1.8203933053466803637371387679$

Torsion generators

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

\( \left(6627, 576286\right) \)  Toggle raw display

Integral points

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

\( \left(3, -2\right) \), \( \left(187, 35326\right) \), \( \left(187, -35514\right) \), \( \left(327, 47086\right) \), \( \left(327, -47414\right) \), \( \left(1027, 89086\right) \), \( \left(1027, -90114\right) \), \( \left(2307, 165886\right) \), \( \left(2307, -168194\right) \), \( \left(2947, 211966\right) \), \( \left(2947, -214914\right) \), \( \left(6627, 576286\right) \), \( \left(6627, -582914\right) \), \( \left(36867, 7077886\right) \), \( \left(36867, -7114754\right) \), \( \left(348177, 205278586\right) \), \( \left(348177, -205626764\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 14490 \)  =  \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-19972835531292672000000 \)  =  \(-1 \cdot 2^{30} \cdot 3^{8} \cdot 5^{6} \cdot 7^{3} \cdot 23^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{47342661265381757089751}{27397579603968000000} \)  =  \(2^{-30} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-3} \cdot 23^{-2} \cdot 37^{3} \cdot 977723^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(2.9685220416860085707008341027\dots\)
Stable Faltings height: \(2.4192158973519537250032114842\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.8203933053466803637371387679\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.072634859145195383930917588467\dots\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 2160 \)  = \( ( 2 \cdot 3 \cdot 5 )\cdot2\cdot( 2 \cdot 3 )\cdot3\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(6\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 14490.2.a.by

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

\( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - 4q^{13} + q^{14} + q^{16} + 2q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1105920
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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

\( L'(E,1) \) ≈ \( 7.9334406793627667527919481919980946712 \)

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(30\) \(I_{30}\) Split multiplicative -1 1 30 30
\(3\) \(2\) \(I_2^{*}\) Additive -1 2 8 2
\(5\) \(6\) \(I_{6}\) Split multiplicative -1 1 6 6
\(7\) \(3\) \(I_{3}\) Split multiplicative -1 1 3 3
\(23\) \(2\) \(I_{2}\) Non-split multiplicative 1 1 2 2

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 X6.

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

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

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.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split add split split ss ordinary ss ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 5 - 2 2 1,1 1 3,1 3 1 1 1 1 1 1 1
$\mu$-invariant(s) 0 - 0 0 0,0 0 0,0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 14490.by consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-7}) \) \(\Z/2\Z \times \Z/6\Z\) Not in database
$4$ 4.2.2132928.1 \(\Z/12\Z\) Not in database
$6$ 6.0.49572993627.2 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.0.222919710806016.3 \(\Z/2\Z \times \Z/12\Z\) Not in database
$9$ 9.3.472409263305056763000000.2 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.0.76547485635354272438168481561649091967000000000000.1 \(\Z/2\Z \times \Z/18\Z\) 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.