# Properties

 Label 55545.s1 Conductor 55545 Discriminant -1425546504621186913546875 j-invariant $$\frac{2744564518708084736}{9629735831296875}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, 15430225, 52499028281]); // or

magma: E := EllipticCurve("55545y1");

sage: E = EllipticCurve([0, 1, 1, 15430225, 52499028281]) # or

sage: E = EllipticCurve("55545y1")

gp: E = ellinit([0, 1, 1, 15430225, 52499028281]) \\ or

gp: E = ellinit("55545y1")

$$y^2 + y = x^{3} + x^{2} + 15430225 x + 52499028281$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-905, -194408\right)$$ $$\hat{h}(P)$$ ≈ 0.0712116481724

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-2375, 49612\right)$$, $$\left(-2375, -49613\right)$$, $$\left(-905, 194407\right)$$, $$\left(-905, -194408\right)$$, $$\left(3925, 416587\right)$$, $$\left(3925, -416588\right)$$, $$\left(30835, 5463247\right)$$, $$\left(30835, -5463248\right)$$, $$\left(111025, 37018012\right)$$, $$\left(111025, -37018013\right)$$

## 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: $$-1425546504621186913546875$$ = $$-1 \cdot 3^{13} \cdot 5^{6} \cdot 7^{5} \cdot 23^{7}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{2744564518708084736}{9629735831296875}$$ = $$2^{15} \cdot 3^{-13} \cdot 5^{-6} \cdot 7^{-5} \cdot 23^{-1} \cdot 43753^{3}$$ 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: $$0.0712116481724$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0604828096516$$ 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: $$1560$$  = $$13\cdot( 2 \cdot 3 )\cdot5\cdot2^{2}$$ 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.s

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 4942080 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$6.71904567577$$

## 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$$(j)_{-}$$
$$3$$ $$13$$ $$I_{13}$$ Split multiplicative -1 1 13 13
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$7$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$23$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1

## 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 split ordinary ss ordinary ordinary add ordinary ss ordinary ordinary ordinary ordinary 3,4 2 6 2 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 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.s 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 3.1.1932.1 $$\Z/2\Z$$ Not in database
6 6.0.1802857392.1 $$\Z/2\Z \times \Z/2\Z$$ 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.