Properties

Label 10008.h1
Conductor 10008
Discriminant -680788675584
j-invariant \( -\frac{82013318212}{911979} \)
CM no
Rank 0
Torsion Structure \(\mathrm{Trivial}\)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -8211, 289118]); // or
magma: E := EllipticCurve("10008h1");
sage: E = EllipticCurve([0, 0, 0, -8211, 289118]) # or
sage: E = EllipticCurve("10008h1")
gp: E = ellinit([0, 0, 0, -8211, 289118]) \\ or
gp: E = ellinit("10008h1")

\( y^2 = x^{3} - 8211 x + 289118 \)

Mordell-Weil group structure

Trivial

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 10008 \)  =  \(2^{3} \cdot 3^{2} \cdot 139\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-680788675584 \)  =  \(-1 \cdot 2^{10} \cdot 3^{14} \cdot 139 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( -\frac{82013318212}{911979} \)  =  \(-1 \cdot 2^{2} \cdot 3^{-8} \cdot 7^{3} \cdot 17^{3} \cdot 23^{3} \cdot 139^{-1}\)
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: \(0.910559829875\)
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: \( 4 \)  = \( 2\cdot2\cdot1 \)
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 10008.2.a.h

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)

\( q + 3q^{5} + 5q^{7} - 5q^{11} - q^{13} + 8q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 26624
\( \Gamma_0(N) \)-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])

\( L(E,1) \) ≈ \( 3.6422393195 \)

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\((j)_{-}\)
\(2\) \(2\) \( III^{*} \) Additive -1 3 10 0
\(3\) \(2\) \( I_8^{*} \) Additive -1 2 14 8
\(139\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 139
Reduction type add add ordinary ordinary ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary split
$\lambda$-invariant(s) - - 0 0 0 0 0,0 0 0,0 2 0 0 0 0 0 1
$\mu$-invariant(s) - - 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 10008.h 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 3.1.139.1 \(\Z/2\Z\) Not in database
6 6.0.2685619.1 \(\Z/2\Z \times \Z/2\Z\) 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.