Properties

 Label 2310g7 Conductor 2310 Discriminant 7641104625000000000 j-invariant $$\frac{1826870018430810435423307849}{7641104625000000000}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

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")

magma: E := EllipticCurve([1, 0, 1, -25467989, -49471735264]); // or

magma: E := EllipticCurve("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

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-\frac{1165779}{400}, \frac{14993321}{8000}\right)$$ $$\hat{h}(P)$$ ≈ 9.982572605409093

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

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

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2310$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$7641104625000000000$$ = $$2^{9} \cdot 3^{8} \cdot 5^{12} \cdot 7 \cdot 11^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); 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

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$9.98257260541$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0672049097647$$ 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: $$16$$  = $$1\cdot2^{3}\cdot2\cdot1\cdot1$$ 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

Modular form2310.2.a.h

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 165888 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$2.68351156466$$

Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{9}$$ 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.

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
$$3$$ 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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 nonsplit split nonsplit split nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss 1 2 1 2 1 1 1 1 1,1 1 1 1 1 1 1,1 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 2310g 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 $$\Q(\sqrt{-3})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{-2})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-77})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{154})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
3 3.1.11907.1 $$\Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{-77})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-2}, \sqrt{-77})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{154})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6 6.0.425329947.3 $$\Z/3\Z \times \Z/6\Z$$ Not in database
6.0.72589644288.18 $$\Z/12\Z$$ Not in database
$$x^{6} - 670824$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$x^{6} - 126 x^{3} + 6237 x^{2} - 45738 x + 87822$$ $$\Z/12\Z$$ 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.