# Properties

 Label 2310.h6 Conductor 2310 Discriminant 56330588160 j-invariant $$\frac{2601656892010848045529}{56330588160}$$ CM no Rank 1 Torsion Structure $$\Z/{6}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -286534, 59011352]) # or

sage: E = EllipticCurve("2310g1")

gp: E = ellinit([1, 0, 1, -286534, 59011352]) \\ or

gp: E = ellinit("2310g1")

magma: E := EllipticCurve([1, 0, 1, -286534, 59011352]); // or

magma: E := EllipticCurve("2310g1");

$$y^2 + x y + y = x^{3} - 286534 x + 59011352$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(334, 605\right)$$ $$\hat{h}(P)$$ ≈ 3.327524201803031

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(373, 1829\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(309, -155\right)$$, $$\left(310, -124\right)$$, $$\left(310, -187\right)$$, $$\left(334, 605\right)$$, $$\left(334, -940\right)$$, $$\left(373, 1829\right)$$, $$\left(373, -2203\right)$$, $$\left(478, 5357\right)$$, $$\left(478, -5836\right)$$, $$\left(4405, 288101\right)$$, $$\left(4405, -292507\right)$$

## 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: $$56330588160$$ = $$2^{12} \cdot 3^{6} \cdot 5 \cdot 7^{3} \cdot 11$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2601656892010848045529}{56330588160}$$ = $$2^{-12} \cdot 3^{-6} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{-1} \cdot 23^{3} \cdot 61^{3} \cdot 9803^{3}$$ 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: $$3.3275242018$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.806458917176$$ 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: $$36$$  = $$2\cdot( 2 \cdot 3 )\cdot1\cdot3\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$6$$ 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: 13824 $$\Gamma_0(N)$$-optimal: yes 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12
$$3$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$11$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## 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 X13c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ 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.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## 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 0 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, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{5})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{77})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{385})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
4 $$\Q(\sqrt{5}, \sqrt{77})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.247066875.1 $$\Z/3\Z \times \Z/6\Z$$ 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.