# Properties

 Label 53312.h2 Conductor $53312$ Discriminant $-5.940\times 10^{16}$ j-invariant $$\frac{63905303}{39304}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, 95583, 2881759]) # or

sage: E = EllipticCurve("53312bk2")

gp: E = ellinit([0, 1, 0, 95583, 2881759]) \\ or

gp: E = ellinit("53312bk2")

magma: E := EllipticCurve([0, 1, 0, 95583, 2881759]); // or

magma: E := EllipticCurve("53312bk2");

$$y^2 = x^{3} + x^{2} + 95583 x + 2881759$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$53312$$ = $$2^{6} \cdot 7^{2} \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-59396518970392576$$ = $$-1 \cdot 2^{21} \cdot 7^{8} \cdot 17^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{63905303}{39304}$$ = $$2^{-3} \cdot 7 \cdot 11^{3} \cdot 17^{-3} \cdot 19^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.216821993458$$ 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: $$4$$  = $$2^{2}\cdot1\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 53312.2.a.h

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);

$$q - 2q^{3} - 3q^{5} + q^{9} - 2q^{13} + 6q^{15} - q^{17} - 7q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 483840 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$0.867287973831$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_11^{*}$$ Additive -1 6 21 3
$$7$$ $$1$$ $$IV^{*}$$ Additive 1 2 8 0
$$17$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3

## Galois representations

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

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 $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B

## $p$-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 add ordinary ordinary add ss ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss - 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 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 $$d=$$ 3.
Its isogeny class 53312.h consists of 2 curves linked by isogenies of degree 3.

## 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
$2$ $$\Q(\sqrt{6})$$ $$\Z/3\Z$$ Not in database
$3$ 3.1.6664.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.896168448.4 $$\Z/3\Z$$ Not in database
$6$ 6.0.6039609856.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database
$6$ 6.2.9592321536.1 $$\Z/6\Z$$ 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.