# Properties

 Label 2310b2 Conductor $2310$ Discriminant $1.405\times 10^{18}$ j-invariant $$\frac{2436531580079063806249}{1405478914998681600}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3+x^2-280338x-3362508$$ y^2+xy=x^3+x^2-280338x-3362508 (homogenize, simplify) $$y^2z+xyz=x^3+x^2z-280338xz^2-3362508z^3$$ y^2z+xyz=x^3+x^2z-280338xz^2-3362508z^3 (dehomogenize, simplify) $$y^2=x^3-363318723x-151431395778$$ y^2=x^3-363318723x-151431395778 (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 0, -280338, -3362508])

gp: E = ellinit([1, 1, 0, -280338, -3362508])

magma: E := EllipticCurve([1, 1, 0, -280338, -3362508]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-524, 262\right)$$, $$\left(-12, 6\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-524, 262\right)$$, $$\left(-12, 6\right)$$

## 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: $1405478914998681600$ = $2^{14} \cdot 3^{14} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2436531580079063806249}{1405478914998681600}$$ = $2^{-14} \cdot 3^{-14} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-4} \cdot 13456249^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.1717826049468938850484970020\dots$ Stable Faltings height: $2.1717826049468938850484970020\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.22624507309304518937687448179\dots$ 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: $64$  = $2\cdot2\cdot2\cdot2\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) 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); Special value: $L(E,1)$ ≈ $0.90498029237218075750749792714$

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 50176 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is semistable. There are 5 primes of bad reduction:

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$(j)_{-}$
$2$ $2$ $I_{14}$ Non-split multiplicative 1 1 14 14
$3$ $2$ $I_{14}$ Non-split multiplicative 1 1 14 14
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.12.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 nonsplit nonsplit nonsplit split 3 0 0 0 5 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 2310b consists of 4 curves linked by isogenies of degrees dividing 4.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{3}, \sqrt{35})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-35})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.62272557540270000.4 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ 16.0.63456228123711897600000000.10 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.