Properties

 Label 1008l6 Conductor $1008$ Discriminant $-1.260\times 10^{16}$ j-invariant $$\frac{6359387729183}{4218578658}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 55581, 1929058]) # or

sage: E = EllipticCurve("1008.j6")

gp: E = ellinit([0, 0, 0, 55581, 1929058]) \\ or

gp: E = ellinit("1008.j6")

magma: E := EllipticCurve([0, 0, 0, 55581, 1929058]); // or

magma: E := EllipticCurve("1008.j6");

$$y^2=x^3+55581x+1929058$$

Mordell-Weil group structure

$$\Z/{2}\Z$$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

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

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

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

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1008$$ = $$2^{4} \cdot 3^{2} \cdot 7$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-12596608375529472$$ = $$-1 \cdot 2^{13} \cdot 3^{22} \cdot 7^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{6359387729183}{4218578658}$$ = $$2^{-1} \cdot 3^{-16} \cdot 7^{-2} \cdot 97^{3} \cdot 191^{3}$$ 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); 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: $$0.25081881564252364911352739139$$ 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: $$32$$  = $$2^{2}\cdot2^{2}\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ 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);

$$q + 2q^{5} + q^{7} - 4q^{11} + 6q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6144 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$2.0065505251401891929082191311383116601$$

Local data

This elliptic curve is semistable. There are 3 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_5^{*}$$ Additive -1 4 13 1
$$3$$ $$4$$ $$I_{16}^{*}$$ Additive -1 2 22 16
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X217.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right)$ and has index 48.

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
$$2$$ 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) 2 3 7 add add split - - 1 - - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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, 4 and 8.
Its isogeny class 1008l consists of 4 curves linked by isogenies of degrees dividing 8.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-2})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{6})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-3})$$ $$\Z/4\Z$$ 2.0.3.1-37632.2-r2 $4$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-14})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{7})$$ $$\Z/8\Z$$ Not in database $8$ 8.0.2498119335936.33 $$\Z/16\Z$$ Not in database $8$ 8.0.3262849744896.31 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.1358954496.2 $$\Z/8\Z$$ Not in database $8$ 8.0.12745506816.5 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.435537865728.5 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ 16.0.1846757322198614016.5 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.