Properties

 Label 2310g2 Conductor 2310 Discriminant 12104550027662400 j-invariant $$\frac{2610383204210122997209}{12104550027662400}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{6}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -286854, 58872856]); // or
magma: E := EllipticCurve("2310g2");
sage: E = EllipticCurve([1, 0, 1, -286854, 58872856]) # or
sage: E = EllipticCurve("2310g2")
gp: E = ellinit([1, 0, 1, -286854, 58872856]) \\ or
gp: E = ellinit("2310g2")

$$y^2 + x y + y = x^{3} - 286854 x + 58872856$$

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()

 $$P$$ = $$\left(-571, 6333\right)$$ $$\hat{h}(P)$$ ≈ 1.6637621009

Torsion generators

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

$$\left(293, -147\right)$$, $$\left(941, 24477\right)$$

Integral points

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

$$\left(-571, 6333\right)$$, $$\left(-214, 10617\right)$$, $$\left(-67, 8853\right)$$, $$\left(50, 6657\right)$$, $$\left(248, 1608\right)$$, $$\left(269, 957\right)$$, $$\left(293, -147\right)$$, $$\left(325, -163\right)$$, $$\left(374, 1797\right)$$, $$\left(413, 3093\right)$$, $$\left(668, 12528\right)$$, $$\left(941, 24477\right)$$, $$\left(1508, 54528\right)$$, $$\left(17573, 2319693\right)$$

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

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: $$12104550027662400$$ = $$2^{6} \cdot 3^{12} \cdot 5^{2} \cdot 7^{6} \cdot 11^{2}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{2610383204210122997209}{12104550027662400}$$ = $$2^{-6} \cdot 3^{-12} \cdot 5^{-2} \cdot 7^{-6} \cdot 11^{-2} \cdot 67^{3} \cdot 205507^{3}$$ 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: $$1.6637621009$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.403229458588$$ 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: $$576$$  = $$2\cdot( 2^{2} \cdot 3 )\cdot2\cdot( 2 \cdot 3 )\cdot2$$ 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 form2310.2.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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 27648 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$2.68351156466$$

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$11$$ $$2$$ $$I_{2}$$ Non-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 X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ 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
$$2$$ 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

 $p$ 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 1 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 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-2}, \sqrt{5})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{77})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-5}, \sqrt{-77})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.247066875.1 $$\Z/6\Z \times \Z/6\Z$$ 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.