Properties

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

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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]);
 

\(y^2=x^3+x^2-2381x-123681\)  Toggle raw display

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

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

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\((66,\pm 105)\)  Toggle raw display

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

Modular form   4620.2.a.k

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);
 

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

For more coefficients, see the Downloads section to the right.

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add split nonsplit split split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ss
$\lambda$-invariant(s) - 2 1 2 2 1 1 1 3 1 1 1 1,1 1 1,1
$\mu$-invariant(s) - 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) 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.