Properties

Label 14490bi1
Conductor $14490$
Discriminant $6.315\times 10^{21}$
j-invariant \( \frac{489781415227546051766883}{233890092903563264000} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -4926602, -1758530871])
 
gp: E = ellinit([1, -1, 1, -4926602, -1758530871])
 
magma: E := EllipticCurve([1, -1, 1, -4926602, -1758530871]);
 

\(y^2+xy+y=x^3-x^2-4926602x-1758530871\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(2577, 50231\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.10958177108665161328996861631$

Torsion generators

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

\( \left(-367, 183\right) \)  Toggle raw display

Integral points

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

\( \left(-1903, 27831\right) \), \( \left(-1903, -25929\right) \), \( \left(-1231, 50007\right) \), \( \left(-1231, -48777\right) \), \( \left(-643, 34131\right) \), \( \left(-643, -33489\right) \), \( \left(-373, 5391\right) \), \( \left(-373, -5019\right) \), \( \left(-367, 183\right) \), \( \left(2577, 50231\right) \), \( \left(2577, -52809\right) \), \( \left(4257, 231111\right) \), \( \left(4257, -235369\right) \), \( \left(16017, 1999031\right) \), \( \left(16017, -2015049\right) \), \( \left(21897, 3212271\right) \), \( \left(21897, -3234169\right) \), \( \left(753297, 653426871\right) \), \( \left(753297, -654180169\right) \)  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: $6315032508396208128000 $  =  $2^{32} \cdot 3^{3} \cdot 5^{3} \cdot 7^{7} \cdot 23^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{489781415227546051766883}{233890092903563264000} \)  =  $2^{-32} \cdot 3^{6} \cdot 5^{-3} \cdot 7^{-7} \cdot 23^{-2} \cdot 47^{3} \cdot 307^{3} \cdot 607^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.8769951741602424514298686676\dots$
Stable Faltings height: $2.6023421019932150285810573584\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.10958177108665161328996861631\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.10623074991337307720837098517\dots$
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: $ 2688 $  = $ 2^{5}\cdot2\cdot3\cdot7\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
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);
 
Special value: $ L'(E,1) $ ≈ $ 7.8227208994170326127811196996357651233 $

Modular invariants

Modular form 14490.2.a.ca

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);
 

\( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} + 2q^{11} - 6q^{13} + q^{14} + q^{16} - 6q^{17} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1032192
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

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$(j)_{-}$
$2$ $32$ $I_{32}$ Split multiplicative -1 1 32 32
$3$ $2$ $III$ Additive 1 2 3 0
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$23$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1

$p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split add split split ordinary ordinary ordinary ss split ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 3 - 2 4 1 1 1 1,1 2 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.
Its isogeny class 14490bi consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{105}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.0.7998480.4 \(\Z/4\Z\) Not in database
$8$ 8.2.23511063249072.8 \(\Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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.