Properties

Label 53130.cs3
Conductor 53130
Discriminant 46261712772301395600
j-invariant \( \frac{381872841673941838282574504161}{46261712772301395600} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -151146590, -715242391500]); // or
magma: E := EllipticCurve("53130cr6");
sage: E = EllipticCurve([1, 0, 0, -151146590, -715242391500]) # or
sage: E = EllipticCurve("53130cr6")
gp: E = ellinit([1, 0, 0, -151146590, -715242391500]) \\ or
gp: E = ellinit("53130cr6")

\( y^2 + x y = x^{3} - 151146590 x - 715242391500 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-7098, 3276\right) \)
\(\hat{h}(P)\) ≈  4.1511527705

Torsion generators

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

\( \left(-7100, 3550\right) \), \( \left(14196, -7098\right) \)

Integral points

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

\( \left(-7100, 3550\right) \), \( \left(-7098, 3822\right) \), \( \left(14196, -7098\right) \), \( \left(32830, 5434030\right) \), \( \left(38530, 7098430\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 53130 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(46261712772301395600 \)  =  \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{6} \cdot 23^{6} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{381872841673941838282574504161}{46261712772301395600} \)  =  \(2^{-4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-6} \cdot 13^{6} \cdot 23^{-6} \cdot 457^{3} \cdot 93937^{3}\)
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: \(4.1511527705\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.0430576149245\)
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: \( 128 \)  = \( 2^{2}\cdot2\cdot2\cdot2\cdot2\cdot2 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(4\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(9\) (exact)

Modular invariants

Modular form 53130.2.a.cs

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)

\( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} + 2q^{13} + q^{14} + q^{15} + q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 7962624
\( \Gamma_0(N) \)-optimal: no
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])

\( L'(E,1) \) ≈ \( 12.8691890989 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(3\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(5\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(7\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(11\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(23\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6

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.1.2

$p$-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 split split split split nonsplit ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 5 4 2 2 1 1 1 1 1 1 1 1 1 1 1,1
$\mu$-invariant(s) 0 1 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 53130.cs 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-3}) \) \(\Z/2\Z \times \Z/6\Z\) Not in database
3 3.1.1190700.7 \(\Z/2\Z \times \Z/6\Z\) Not in database
4 \(\Q(\sqrt{-11}, \sqrt{-15})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{11}, \sqrt{161})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{15}, \sqrt{-161})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
6 6.0.4253299470000.8 \(\Z/6\Z \times \Z/6\Z\) 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.