Properties

 Label 5712.bb1 Conductor $5712$ Discriminant $-1.407\times 10^{14}$ j-invariant $$-\frac{1184052061112257}{34349180544}$$ 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, -35264, 2600244])

gp: E = ellinit([0, 1, 0, -35264, 2600244])

magma: E := EllipticCurve([0, 1, 0, -35264, 2600244]);

$$y^2=x^3+x^2-35264x+2600244$$

trivial

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);



Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$5712$$ = $$2^{4} \cdot 3 \cdot 7 \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-140694243508224$$ = $$-1 \cdot 2^{19} \cdot 3^{3} \cdot 7 \cdot 17^{5}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1184052061112257}{34349180544}$$ = $$-1 \cdot 2^{-7} \cdot 3^{-3} \cdot 7^{-1} \cdot 17^{-5} \cdot 67^{3} \cdot 1579^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.4940154263734649692321238088\dots$$ Stable Faltings height: $$0.80086824581351965981489168734\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.57949736298433970687921485381\dots$$ 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: $$6$$  = $$2\cdot3\cdot1\cdot1$$ 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

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);

$$q + q^{3} + 3q^{5} + q^{7} + q^{9} - q^{11} + q^{13} + 3q^{15} - q^{17} - 6q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 20160 $$\Gamma_0(N)$$-optimal: yes 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);

$$L(E,1)$$ ≈ $$3.4769841779060382412752891228794195994$$

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{11}^{*}$$ Additive -1 4 19 7
$$3$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$7$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$17$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5

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 $$p$$ .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 split ordinary split ordinary ordinary nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary ss - 1 0 3 0 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

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has no rational isogenies. Its isogeny class 5712.bb consists of this curve only.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.2856.1 $$\Z/2\Z$$ Not in database $6$ 6.0.23295638016.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.