Properties

Label 2310.h1
Conductor 2310
Discriminant 7641104625000000000
j-invariant \( \frac{1826870018430810435423307849}{7641104625000000000} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -25467989, -49471735264]); // or
magma: E := EllipticCurve("2310g7");
sage: E = EllipticCurve([1, 0, 1, -25467989, -49471735264]) # or
sage: E = EllipticCurve("2310g7")
gp: E = ellinit([1, 0, 1, -25467989, -49471735264]) \\ or
gp: E = ellinit("2310g7")

\( y^2 + x y + y = x^{3} - 25467989 x - 49471735264 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-\frac{1165779}{400}, \frac{14993321}{8000}\right) \)
\(\hat{h}(P)\) ≈  9.98257260541

Torsion generators

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

\( \left(-\frac{11673}{4}, \frac{11669}{8}\right) \)

Integral points

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 2310 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(7641104625000000000 \)  =  \(2^{9} \cdot 3^{8} \cdot 5^{12} \cdot 7 \cdot 11^{3} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{1826870018430810435423307849}{7641104625000000000} \)  =  \(2^{-9} \cdot 3^{-8} \cdot 5^{-12} \cdot 7^{-1} \cdot 11^{-3} \cdot 109^{3} \cdot 601^{3} \cdot 18661^{3}\)
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: \(9.98257260541\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.0672049097647\)
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: \( 16 \)  = \( 1\cdot2^{3}\cdot2\cdot1\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 2310.2.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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 165888
\( \Gamma_0(N) \)-optimal: no
Manin constant: not computed

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])

\( L'(E,1) \) ≈ \( 2.68351156466 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{9} \) Non-split multiplicative 1 1 9 9
\(3\) \(8\) \( I_{8} \) Split multiplicative -1 1 8 8
\(5\) \(2\) \( I_{12} \) Non-split multiplicative 1 1 12 12
\(7\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(11\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3

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 X13g.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

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
\(2\) B
\(3\) 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 nonsplit split nonsplit split nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 1 2 1 2 1 1 1 1 1,1 1 1 1 1 1 1,1
$\mu$-invariant(s) 2 1 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, 4, 6 and 12.
Its isogeny class 2310.h 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{154}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-77}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{-2}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{-3}) \) \(\Z/6\Z\) Not in database
3 3.1.11907.1 \(\Z/6\Z\) Not in database
4 \(\Q(\sqrt{-2}, \sqrt{-77})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-2}, \sqrt{-3})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{-3}, \sqrt{154})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
\(\Q(\sqrt{-3}, \sqrt{-77})\) \(\Z/12\Z\) Not in database
6 \(x^{6} \) \(\mathstrut -\mathstrut 670824 \) \(\Z/2\Z \times \Z/6\Z\) Not in database
6.0.425329947.3 \(\Z/3\Z \times \Z/6\Z\) Not in database
6.0.72589644288.18 \(\Z/12\Z\) Not in database
\(x^{6} \) \(\mathstrut -\mathstrut 126 x^{3} \) \(\mathstrut +\mathstrut 6237 x^{2} \) \(\mathstrut -\mathstrut 45738 x \) \(\mathstrut +\mathstrut 87822 \) \(\Z/12\Z\) 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.