# Properties

 Label 1110.b1 Conductor $1110$ Discriminant $-3330$ j-invariant $$\frac{357911}{3330}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, 2, -2])

gp: E = ellinit([1, 1, 0, 2, -2])

magma: E := EllipticCurve([1, 1, 0, 2, -2]);

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

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(1, 1\right)$$ $\hat{h}(P)$ ≈ $0.36946917117534265571177776083$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1, 1\right)$$, $$\left(1, -2\right)$$, $$\left(3, 5\right)$$, $$\left(3, -8\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1110$$ = $2 \cdot 3 \cdot 5 \cdot 37$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-3330$ = $-1 \cdot 2 \cdot 3^{2} \cdot 5 \cdot 37$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{357911}{3330}$$ = $2^{-1} \cdot 3^{-2} \cdot 5^{-1} \cdot 37^{-1} \cdot 71^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.64288599121620077661819375319\dots$ Stable Faltings height: $-0.64288599121620077661819375319\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.36946917117534265571177776083\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.2084766971799291391673585513\dots$ 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: $2$  = $1\cdot2\cdot1\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L'(E,1)$ ≈ $1.6319281097342532520960239611072340032$

## Modular invariants

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);

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - q^{11} - q^{12} + 2q^{13} - q^{14} + q^{15} + q^{16} - 7q^{17} - q^{18} - 2q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 80 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is semistable. There are 4 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$(j)_{-}$
$2$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$37$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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 for all primes $\ell$.

## $p$-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 nonsplit nonsplit nonsplit ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary nonsplit ordinary ordinary ordinary 11 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

## Isogenies

This curve has no rational isogenies. Its isogeny class 1110.b consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.1480.1 $$\Z/2\Z$$ Not in database $6$ 6.0.3241792000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.3320019986670000.6 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.