Properties

 Label 18496.j3 Conductor $18496$ Discriminant $98867482624$ j-invariant $$1728$$ CM yes ($$D=-4$$) Rank $2$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -1156, 0]) # or

sage: E = EllipticCurve("18496i2")

gp: E = ellinit([0, 0, 0, -1156, 0]) \\ or

gp: E = ellinit("18496i2")

magma: E := EllipticCurve([0, 0, 0, -1156, 0]); // or

magma: E := EllipticCurve("18496i2");

$$y^2 = x^{3} - 1156 x$$

Mordell-Weil group structure

$$\Z^2 \times \Z/{2}\Z \times \Z/{2}\Z$$

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-16, 120\right)$$ $$\left(-2, 48\right)$$ $$\hat{h}(P)$$ ≈ $3.018669916974929$ $2.5129601807581823$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(0, 0\right)$$, $$\left(34, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-34, 0\right)$$, $$(-16,\pm 120)$$, $$(-2,\pm 48)$$, $$\left(0, 0\right)$$, $$\left(34, 0\right)$$, $$(162,\pm 2016)$$, $$(578,\pm 13872)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$18496$$ = $$2^{6} \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$98867482624$$ = $$2^{12} \cdot 17^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$1728$$ = $$2^{6} \cdot 3^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-1}]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$7.0996751824$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.899358321447$$ 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: $$16$$  = $$2^{2}\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 18496.2.a.j

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);

$$q - 2q^{5} - 3q^{9} - 6q^{13} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 10240 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$6.38515195486$$

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_2^{*}$$ Additive -1 6 12 0
$$17$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0

Galois representations

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 for all primes $$p < 1000$$ except those listed.

prime Image of Galois representation
$$2$$ Cs

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -1 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -1 }{p}\right)=-1$$.

$p$-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 add ss ordinary ss ss ordinary add ss ss ordinary ss ordinary ordinary ss ss - 2,4 2 2,2 2,2 2 - 4,2 2,2 2 2,2 2 2 2,2 2,2 - 0,0 0 0,0 0,0 0 - 0,0 0,0 0 0,0 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 $$d=$$ 2.
Its isogeny class 18496.j consists of 4 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$4$ $$\Q(\sqrt{2}, \sqrt{-17})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(i, \sqrt{34})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{2}, \sqrt{17})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.