# Properties

 Label 49098.b2 Conductor 49098 Discriminant -11316892608 j-invariant $$-\frac{10218313}{96192}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -221, 5181]) # or

sage: E = EllipticCurve("49098e1")

gp: E = ellinit([1, 1, 0, -221, 5181]) \\ or

gp: E = ellinit("49098e1")

magma: E := EllipticCurve([1, 1, 0, -221, 5181]); // or

magma: E := EllipticCurve("49098e1");

$$y^2 + x y = x^{3} + x^{2} - 221 x + 5181$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-15, 81\right)$$ $$\left(-14, 83\right)$$ $$\hat{h}(P)$$ ≈ 1.5157052601599739 2.4111610305872713

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-22, 11\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-22, 11\right)$$, $$\left(-15, 81\right)$$, $$\left(-15, -66\right)$$, $$\left(-14, 83\right)$$, $$\left(-14, -69\right)$$, $$\left(-1, 74\right)$$, $$\left(-1, -73\right)$$, $$\left(3, 66\right)$$, $$\left(3, -69\right)$$, $$\left(10, 59\right)$$, $$\left(10, -69\right)$$, $$\left(34, 179\right)$$, $$\left(34, -213\right)$$, $$\left(125, 1334\right)$$, $$\left(125, -1459\right)$$, $$\left(146, 1691\right)$$, $$\left(146, -1837\right)$$, $$\left(22378, 3336491\right)$$, $$\left(22378, -3358869\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$49098$$ = $$2 \cdot 3 \cdot 7^{2} \cdot 167$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-11316892608$$ = $$-1 \cdot 2^{6} \cdot 3^{2} \cdot 7^{6} \cdot 167$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{10218313}{96192}$$ = $$-1 \cdot 2^{-6} \cdot 3^{-2} \cdot 7^{3} \cdot 31^{3} \cdot 167^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.24624076067$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.08953375455$$ 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: $$16$$  = $$2\cdot2\cdot2^{2}\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

#### Modular form 49098.2.a.b

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);

$$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - q^{8} + q^{9} + 2q^{10} - 4q^{11} - q^{12} + 2q^{15} + q^{16} + 4q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 41472 $$\Gamma_0(N)$$-optimal: yes 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);

$$L^{(2)}(E,1)/2!$$ ≈ $$5.43128550019$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0
$$167$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## 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

## $p$-adic data

### $p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 47 167 nonsplit nonsplit ordinary add ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss nonsplit 5 2 2 - 2 2,2 2 2 2 4 2 2 2 2 2,2 2 0 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 49098.b consists of 2 curves linked by isogenies of degree 2.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-167})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.2.4713408.2 $$\Z/4\Z$$ 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.