# Properties

 Label 2310.o1 Conductor $2310$ Discriminant $9.643\times 10^{13}$ j-invariant $$\frac{78519570041710065450485106721}{96428056919040}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3+x^2-89210930x+324283935935$$ y^2+xy+y=x^3+x^2-89210930x+324283935935 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3+x^2z-89210930xz^2+324283935935z^3$$ y^2z+xyz+yz^2=x^3+x^2z-89210930xz^2+324283935935z^3 (dehomogenize, simplify) $$y^2=x^3-115617365307x+15131525575471254$$ y^2=x^3-115617365307x+15131525575471254 (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 1, -89210930, 324283935935])

gp: E = ellinit([1, 1, 1, -89210930, 324283935935])

magma: E := EllipticCurve([1, 1, 1, -89210930, 324283935935]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{4}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(5465, -1165\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(5465, -1165\right)$$, $$\left(5465, -4301\right)$$

## 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: $96428056919040$ = $2^{10} \cdot 3^{3} \cdot 5 \cdot 7^{8} \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{78519570041710065450485106721}{96428056919040}$$ = $2^{-10} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-8} \cdot 11^{-2} \cdot 2543^{3} \cdot 1683887^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.8574319917522168096962478935\dots$ Stable Faltings height: $2.8574319917522168096962478935\dots$

## 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: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.26843000563938978863981367056\dots$ 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: $160$  = $( 2 \cdot 5 )\cdot1\cdot1\cdot2^{3}\cdot2$ 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: $L(E,1)$ ≈ $2.6843000563938978863981367056$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 184320 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$3$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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
$2$ 2B 8.48.0.161

The image of the adelic Galois representation has level $1680$, index $192$, and genus $1$.

## $p$-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 split nonsplit split split nonsplit 3 0 3 1 0 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 2310.o consists of 6 curves linked by isogenies of degrees dividing 8.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{15})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{30})$$ $$\Z/8\Z$$ Not in database $2$ $$\Q(\sqrt{2})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{15})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.42693156000000.39 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.10929447936000000.78 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/16\Z$$ Not in database $8$ Deg 8 $$\Z/16\Z$$ Not in database $8$ 8.2.768797006670000.2 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ 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.