# Properties

 Label 2310.l5 Conductor 2310 Discriminant 270140062500 j-invariant $$\frac{432288716775559561}{270140062500}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z \times \Z/{6}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -15753, 759256]) # or

sage: E = EllipticCurve("2310l2")

gp: E = ellinit([1, 0, 1, -15753, 759256]) \\ or

gp: E = ellinit("2310l2")

magma: E := EllipticCurve([1, 0, 1, -15753, 759256]); // or

magma: E := EllipticCurve("2310l2");

$$y^2 + x y + y = x^{3} - 15753 x + 759256$$

## Mordell-Weil group structure

$$\Z/{2}\Z \times \Z/{6}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(71, -36\right)$$, $$\left(170, 1647\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-145, 72\right)$$, $$\left(5, 822\right)$$, $$\left(5, -828\right)$$, $$\left(65, 72\right)$$, $$\left(65, -138\right)$$, $$\left(71, -36\right)$$, $$\left(80, 72\right)$$, $$\left(80, -153\right)$$, $$\left(170, 1647\right)$$, $$\left(170, -1818\right)$$

## 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: $$270140062500$$ = $$2^{2} \cdot 3^{6} \cdot 5^{6} \cdot 7^{2} \cdot 11^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{432288716775559561}{270140062500}$$ = $$2^{-2} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-2} \cdot 31^{3} \cdot 24391^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.968912908708$$ 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: $$288$$  = $$2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$12$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form2310.2.a.l

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: 4608 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$1.93782581742$$

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

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
$$2$$ 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]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 nonsplit split split split nonsplit 3 1 3 1 0 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 2310.l 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 $$\Q(\sqrt{6}, \sqrt{-11})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-6}, \sqrt{-35})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{11}, \sqrt{35})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.15186113712.1 $$\Z/6\Z \times \Z/6\Z$$ 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.