# Properties

 Label 2310.l4 Conductor 2310 Discriminant 757333463040 j-invariant $$\frac{12038605770121350841}{757333463040}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -47748, -4019582]); // or

magma: E := EllipticCurve("2310l3");

sage: E = EllipticCurve([1, 0, 1, -47748, -4019582]) # or

sage: E = EllipticCurve("2310l3")

gp: E = ellinit([1, 0, 1, -47748, -4019582]) \\ or

gp: E = ellinit("2310l3")

$$y^2 + x y + y = x^{3} - 47748 x - 4019582$$

## Mordell-Weil group structure

$$\Z/{2}\Z$$

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-127, 63\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-127, 63\right)$$

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: $$757333463040$$ = $$2^{12} \cdot 3^{4} \cdot 5 \cdot 7^{3} \cdot 11^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{12038605770121350841}{757333463040}$$ = $$2^{-12} \cdot 3^{-4} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{-3} \cdot 23^{3} \cdot 251^{3} \cdot 397^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$0$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.322970969569$$ 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: $$24$$  = $$2\cdot2^{2}\cdot1\cdot3\cdot1$$ 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 form2310.2.a.l

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: 6912 $$\Gamma_0(N)$$-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])

$$L(E,1)$$ ≈ $$1.93782581742$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12
$$3$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$11$$ $$1$$ $$I_{3}$$ 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 X13.

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

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 1 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, 4, 6 and 12.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{385})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{-11})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-35})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-3})$$ $$\Z/6\Z$$ Not in database
3 3.1.6075.2 $$\Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-3}, \sqrt{-35})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-11}, \sqrt{-35})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{-11})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{385})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6 6.0.110716875.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database
$$x^{6} - 3 x^{5} - 285 x^{4} + 545 x^{3} + 27405 x^{2} - 36333 x - 880176$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.0.49121386875.3 $$\Z/12\Z$$ Not in database
6.0.63293146875.5 $$\Z/12\Z$$ 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.