# Properties

 Label 55545.y1 Conductor 55545 Discriminant -5315614178875875 j-invariant $$-\frac{18163305455448064}{10048419997875}$$ CM no Rank 0 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -44290, -5032331]); // or

magma: E := EllipticCurve("55545v1");

sage: E = EllipticCurve([0, 1, 1, -44290, -5032331]) # or

sage: E = EllipticCurve("55545v1")

gp: E = ellinit([0, 1, 1, -44290, -5032331]) \\ or

gp: E = ellinit("55545v1")

$$y^2 + y = x^{3} + x^{2} - 44290 x - 5032331$$

Trivial

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$55545$$ = $$3 \cdot 5 \cdot 7 \cdot 23^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-5315614178875875$$ = $$-1 \cdot 3^{14} \cdot 5^{3} \cdot 7^{5} \cdot 23^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{18163305455448064}{10048419997875}$$ = $$-1 \cdot 2^{12} \cdot 3^{-14} \cdot 5^{-3} \cdot 7^{-5} \cdot 23 \cdot 53^{3} \cdot 109^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$0$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.160421493761$$ 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: $$42$$  = $$( 2 \cdot 7 )\cdot3\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 55545.2.a.y

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 524160 $$\Gamma_0(N)$$-optimal: yes 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])

$$L(E,1)$$ ≈ $$6.73770273796$$

## Local data

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$$(j)_{-}$$
$$3$$ $$14$$ $$I_{14}$$ Split multiplicative -1 1 14 14
$$5$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$7$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5
$$23$$ $$1$$ $$II$$ Additive -1 2 2 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

## $p$-adic data

### $p$-adic regulators

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

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## 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 ss split split nonsplit ordinary ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary 1,2 3 1 2 0 0 0 0 - 0 0 0 0 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 no rational isogenies. Its isogeny class 55545.y consists of this curve only.

## 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
3 3.1.74060.2 $$\Z/2\Z$$ Not in database
6 6.0.191970926000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.