# Properties

 Label 55545x1 Conductor 55545 Discriminant 531461503125 j-invariant $$\frac{22128056725504}{1004653125}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -4730, 118634]); // or

magma: E := EllipticCurve("55545x1");

sage: E = EllipticCurve([0, 1, 1, -4730, 118634]) # or

sage: E = EllipticCurve("55545x1")

gp: E = ellinit([0, 1, 1, -4730, 118634]) \\ or

gp: E = ellinit("55545x1")

$$y^2 + y = x^{3} + x^{2} - 4730 x + 118634$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-74, 262\right)$$ $$\left(1, 337\right)$$ $$\hat{h}(P)$$ ≈ 0.880828685336 0.192867128043

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-74, 262\right)$$, $$\left(-74, -263\right)$$, $$\left(-59, 442\right)$$, $$\left(-59, -443\right)$$, $$\left(-53, 472\right)$$, $$\left(-53, -473\right)$$, $$\left(1, 337\right)$$, $$\left(1, -338\right)$$, $$\left(16, 217\right)$$, $$\left(16, -218\right)$$, $$\left(28, 94\right)$$, $$\left(28, -95\right)$$, $$\left(31, 52\right)$$, $$\left(31, -53\right)$$, $$\left(46, 22\right)$$, $$\left(46, -23\right)$$, $$\left(51, 112\right)$$, $$\left(51, -113\right)$$, $$\left(73, 409\right)$$, $$\left(73, -410\right)$$, $$\left(136, 1417\right)$$, $$\left(136, -1418\right)$$, $$\left(226, 3262\right)$$, $$\left(226, -3263\right)$$, $$\left(276, 4462\right)$$, $$\left(276, -4463\right)$$, $$\left(631, 15772\right)$$, $$\left(631, -15773\right)$$, $$\left(1351, 49612\right)$$, $$\left(1351, -49613\right)$$, $$\left(2341, 113242\right)$$, $$\left(2341, -113243\right)$$

## 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: $$531461503125$$ = $$3^{8} \cdot 5^{5} \cdot 7^{2} \cdot 23^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{22128056725504}{1004653125}$$ = $$2^{12} \cdot 3^{-8} \cdot 5^{-5} \cdot 7^{-2} \cdot 23 \cdot 617^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.0819555542635$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.915586994656$$ 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: $$80$$  = $$2^{3}\cdot5\cdot2\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

#### Modular form 55545.2.a.b

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 153600 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$6.00299516988$$

## 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$$(j)_{-}$$
$$3$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$5$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$23$$ $$1$$ $$II$$ 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]

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 ss split split nonsplit ordinary ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary 3,6 3 3 2 2 2 2 4 - 2 2 2 2 2 2 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 55545x 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 3.3.10580.1 $$\Z/2\Z$$ Not in database
6 6.6.559682000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.