# Properties

 Label 4620k2 Conductor $4620$ Discriminant $-5.660\times 10^{12}$ j-invariant $$-\frac{5833703071744}{22107421875}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -2381, -123681])

gp: E = ellinit([0, 1, 0, -2381, -123681])

magma: E := EllipticCurve([0, 1, 0, -2381, -123681]);

$$y^2=x^3+x^2-2381x-123681$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(66, 105\right)$$ $$\hat{h}(P)$$ ≈ $4.6351336573425467323594631613$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(66,\pm 105)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$4620$$ = $$2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-5659500000000$$ = $$-1 \cdot 2^{8} \cdot 3 \cdot 5^{9} \cdot 7^{3} \cdot 11$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{5833703071744}{22107421875}$$ = $$-1 \cdot 2^{13} \cdot 3^{-1} \cdot 5^{-9} \cdot 7^{-3} \cdot 11^{-1} \cdot 19^{3} \cdot 47^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.1326299918786166753878078146\dots$$ Stable Faltings height: $$0.67053187150531980244298640029\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$4.6351336573425467323594631613\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.31260133270528141241714076543\dots$$ 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: $$3$$  = $$1\cdot1\cdot1\cdot3\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## 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);

$$q + q^{3} - q^{5} + q^{7} + q^{9} + q^{11} + 2q^{13} - q^{15} - 3q^{17} - 7q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7776 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$4.3468468756571559042017832473820937353$$

## Local data

This elliptic curve is not 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$$(j)_{-}$$
$$2$$ $$1$$ $$IV^{*}$$ Additive -1 2 8 0
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$11$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
$$3$$ B.1.2

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 add split nonsplit split split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ss - 2 1 2 2 1 1 1 3 1 1 1 1,1 1 1,1 - 1 0 0 0 0 0 0 0 0 0 0 0,0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 4620k consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.4620.1 $$\Z/2\Z$$ Not in database $3$ 3.1.117612.2 $$\Z/3\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.41497747632.4 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.64033200.1 $$\Z/6\Z$$ Not in database $9$ 9.1.2301828583191102504000.2 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.7506488190975662418173164216130383872.4 $$\Z/9\Z$$ Not in database $18$ 18.0.15895244479186674903049991493105210048000000.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.7496594677496415551150952237935744688888000000000.1 $$\Z/2\Z \times \Z/6\Z$$ 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.