# Properties

 Label 55470bb2 Conductor $55470$ Discriminant $-133128000000$ j-invariant $$-\frac{337335507529}{72000000}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -1780, 33077]) # or

sage: E = EllipticCurve("55470bb2")

gp: E = ellinit([1, 1, 1, -1780, 33077]) \\ or

gp: E = ellinit("55470bb2")

magma: E := EllipticCurve([1, 1, 1, -1780, 33077]); // or

magma: E := EllipticCurve("55470bb2");

$$y^2 + x y + y = x^{3} + x^{2} - 1780 x + 33077$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-49, 99\right)$$ $$\left(17, -99\right)$$ $$\hat{h}(P)$$ ≈ $3.1428933256454443$ $0.6226548884865857$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-49, 99\right)$$, $$\left(-49, -51\right)$$, $$\left(-43, 201\right)$$, $$\left(-43, -159\right)$$, $$\left(-23, 261\right)$$, $$\left(-23, -239\right)$$, $$\left(5, 153\right)$$, $$\left(5, -159\right)$$, $$\left(7, 141\right)$$, $$\left(7, -149\right)$$, $$\left(17, 81\right)$$, $$\left(17, -99\right)$$, $$\left(27, 61\right)$$, $$\left(27, -89\right)$$, $$\left(37, 121\right)$$, $$\left(37, -159\right)$$, $$\left(45, 193\right)$$, $$\left(45, -239\right)$$, $$\left(77, 561\right)$$, $$\left(77, -639\right)$$, $$\left(227, 3261\right)$$, $$\left(227, -3489\right)$$, $$\left(397, 7681\right)$$, $$\left(397, -8079\right)$$, $$\left(527, 11811\right)$$, $$\left(527, -12339\right)$$, $$\left(1897, 81681\right)$$, $$\left(1897, -83579\right)$$, $$\left(22637, 3394641\right)$$, $$\left(22637, -3417279\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$55470$$ = $$2 \cdot 3 \cdot 5 \cdot 43^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-133128000000$$ = $$-1 \cdot 2^{9} \cdot 3^{2} \cdot 5^{6} \cdot 43^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{337335507529}{72000000}$$ = $$-1 \cdot 2^{-9} \cdot 3^{-2} \cdot 5^{-6} \cdot 43 \cdot 1987^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential 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: $$0.114029360135$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.993851389143$$ 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: $$108$$  = $$3^{2}\cdot2\cdot( 2 \cdot 3 )\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 55470.2.a.y

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 72576 $$\Gamma_0(N)$$-optimal: no 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);

$$L^{(2)}(E,1)/2!$$ ≈ $$12.2394497011$$

## 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$$(j)_{-}$$
$$2$$ $$9$$ $$I_{9}$$ Split multiplicative -1 1 9 9
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$43$$ $$1$$ $$II$$ Additive -1 2 2 0

## 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 X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

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
$$3$$ 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 split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary add ordinary 6 2 3 2 2 2 2 2 2 2 2 2 2 - 2 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 $$d=$$ 3.
Its isogeny class 55470bb consists of 2 curves linked by isogenies of degree 3.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ $$\Q(\sqrt{129})$$ $$\Z/3\Z$$ Not in database
$3$ 3.1.14792.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.1750426112.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database
$6$ 6.2.254030589504.2 $$\Z/6\Z$$ Not in database
$6$ 6.0.8680701550707.3 $$\Z/3\Z$$ 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.