Properties

 Label 55545v1 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 55545v 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.