# Properties

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

# Related objects

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]);

$$y^2+xy+y=x^3-x^2-4926602x-1758530871$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-367, 183\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1903, 27831\right)$$, $$\left(-1903, -25929\right)$$, $$\left(-1231, 50007\right)$$, $$\left(-1231, -48777\right)$$, $$\left(-643, 34131\right)$$, $$\left(-643, -33489\right)$$, $$\left(-373, 5391\right)$$, $$\left(-373, -5019\right)$$, $$\left(-367, 183\right)$$, $$\left(2577, 50231\right)$$, $$\left(2577, -52809\right)$$, $$\left(4257, 231111\right)$$, $$\left(4257, -235369\right)$$, $$\left(16017, 1999031\right)$$, $$\left(16017, -2015049\right)$$, $$\left(21897, 3212271\right)$$, $$\left(21897, -3234169\right)$$, $$\left(753297, 653426871\right)$$, $$\left(753297, -654180169\right)$$

## 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);

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

 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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 split add split split ordinary ordinary ordinary ss split ordinary ordinary ordinary ordinary ordinary ss 3 - 2 4 1 1 1 1,1 2 1 1 1 1 1 1,1 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{105})$$ $$\Z/2\Z \times \Z/2\Z$$ 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.