# Properties

 Label 19110.bq3 Conductor 19110 Discriminant -10576819922457882813378750 j-invariant $$-\frac{1688971789881664420008241}{89901485966373558750}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -121570716, 539085800763]); // or

magma: E := EllipticCurve("19110bo8");

sage: E = EllipticCurve([1, 1, 1, -121570716, 539085800763]) # or

sage: E = EllipticCurve("19110bo8")

gp: E = ellinit([1, 1, 1, -121570716, 539085800763]) \\ or

gp: E = ellinit("19110bo8")

$$y^2 + x y + y = x^{3} + x^{2} - 121570716 x + 539085800763$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(\frac{7981782147}{662596}, -\frac{492500726625033}{539353144}\right)$$ $$\hat{h}(P)$$ ≈ 20.8618788328

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-\frac{51181}{4}, \frac{51177}{8}\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$19110$$ = $$2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-10576819922457882813378750$$ = $$-1 \cdot 2 \cdot 3^{2} \cdot 5^{4} \cdot 7^{9} \cdot 13^{12}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{1688971789881664420008241}{89901485966373558750}$$ = $$-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{-3} \cdot 13^{-12} \cdot 47^{3} \cdot 157^{3} \cdot 16139^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$20.8618788328$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0712678212973$$ 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: $$16$$  = $$1\cdot2\cdot2\cdot2\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 19110.2.a.bq

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 5308416 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$5.94712261034$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$7$$ $$2$$ $$I_3^{*}$$ Additive -1 2 9 3
$$13$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12

## 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 X13g.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ 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

## $p$-adic data

### $p$-adic regulators

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

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

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 split nonsplit nonsplit add ss nonsplit ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss 5 1 1 - 1,1 1 1 1 1,1 1 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, 3, 4, 6 and 12.
Its isogeny class 19110.bq 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-14})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{21})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{7})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-2})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\sqrt{3}, \sqrt{7})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-2}, \sqrt{21})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-6}, \sqrt{-14})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6 6.0.202538070000.7 $$\Z/6\Z$$ 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.