Properties

Label 51870.y6
Conductor 51870
Discriminant 14944558319037792900
j-invariant \( \frac{35872512095393194378249}{14944558319037792900} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z \times \Z/{6}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -687089, -116077264]); // or
magma: E := EllipticCurve("51870bd2");
sage: E = EllipticCurve([1, 0, 1, -687089, -116077264]) # or
sage: E = EllipticCurve("51870bd2")
gp: E = ellinit([1, 0, 1, -687089, -116077264]) \\ or
gp: E = ellinit("51870bd2")

\( y^2 + x y + y = x^{3} - 687089 x - 116077264 \)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z \times \Z/{6}\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-684, 6172\right) \)
\(\hat{h}(P)\) ≈  2.50300378229

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

\( \left(-177, 88\right) \), \( \left(4893, 334708\right) \)

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

\( \left(-684, 6172\right) \), \( \left(-567, 9838\right) \), \( \left(-427, 10188\right) \), \( \left(-420, 10132\right) \), \( \left(-294, 7927\right) \), \( \left(-177, 88\right) \), \( \left(903, -452\right) \), \( \left(993, 12958\right) \), \( \left(1344, 36592\right) \), \( \left(4893, 334708\right) \), \( \left(18543, 2513248\right) \), \( \left(20454, 2912647\right) \), \( \left(8080653, 22966419598\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 51870 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(14944558319037792900 \)  =  \(2^{2} \cdot 3^{6} \cdot 5^{2} \cdot 7^{6} \cdot 13^{6} \cdot 19^{2} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{35872512095393194378249}{14944558319037792900} \)  =  \(2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-6} \cdot 13^{-6} \cdot 19^{-2} \cdot 31^{3} \cdot 127^{3} \cdot 8377^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(1\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(2.50300378229\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.172060467345\)
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: \( 1728 \)  = \( 2\cdot( 2 \cdot 3 )\cdot2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(12\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 51870.2.a.y

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)

\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} - q^{18} + q^{19} + O(q^{20}) \)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
1327104 . This curve is not \( \Gamma_0(N) \)-optimal.

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])

\( L'(E,1) \) ≈ \( 5.16801600656 \)

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\((j)_{-}\)
\(2\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(3\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(5\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(7\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(13\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(19\) \(2\) \( I_{2} \) 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 X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ and has index 6.

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 \( p \) except those listed.

prime Image of Galois representation
\(2\) Cs
\(3\) B.1.1

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit split nonsplit split ss split ordinary split ss ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 2 2 3 4 1,1 4 1 2 1,1 1 1 1 1 1 1,1
$\mu$-invariant(s) 1 0 0 0 0,0 0 0 0 0,0 0 0 0 0 0 0,0

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 51870.y consists of 8 curves linked by isogenies of degrees dividing 12.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 \(\Q(\sqrt{30}, \sqrt{133})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
\(\Q(\sqrt{-13}, \sqrt{-133})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
\(\Q(\sqrt{13}, \sqrt{-30})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
6 6.0.35186670000.1 \(\Z/6\Z \times \Z/6\Z\) 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.