Properties

Label 334620l2
Conductor $334620$
Discriminant $1.077\times 10^{21}$
j-invariant \( \frac{13584145739344}{1195803675} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -4800783, 3728076118])
 
gp: E = ellinit([0, 0, 0, -4800783, 3728076118])
 
magma: E := EllipticCurve([0, 0, 0, -4800783, 3728076118]);
 

\(y^2=x^3-4800783x+3728076118\)  Toggle raw display

Mordell-Weil group structure

$\Z^2 \times \Z/{2}\Z$

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(599, 32670\right)\)  Toggle raw display\(\left(819, 18590\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.61133008023014352823185506179$$1.0553295535278753178349949682$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(962, 0\right) \)  Toggle raw display

Integral points

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

\((-2506,\pm 4590)\), \((-1561,\pm 86130)\), \((-897,\pm 85514)\), \((599,\pm 32670)\), \((819,\pm 18590)\), \((923,\pm 9126)\), \( \left(962, 0\right) \), \((1599,\pm 11830)\), \((2051,\pm 50094)\), \((2678,\pm 100386)\), \((3987,\pm 219010)\), \((6539,\pm 501930)\), \((13283,\pm 1511154)\), \((15119,\pm 1840410)\), \((52871,\pm 12146706)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 334620 \)  =  $2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $1077178040521503148800 $  =  $2^{8} \cdot 3^{9} \cdot 5^{2} \cdot 11^{6} \cdot 13^{6} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{13584145739344}{1195803675} \)  =  $2^{4} \cdot 3^{-3} \cdot 5^{-2} \cdot 11^{-6} \cdot 17^{3} \cdot 557^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.7760288527568376080914619459\dots$
Stable Faltings height: $0.48214990931871752142227419235\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.63442828328062664748219108728\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.15129579942798579166493219241\dots$
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: $ 576 $  = $ 3\cdot2^{2}\cdot2\cdot( 2 \cdot 3 )\cdot2^{2} $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (rounded)
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^{(2)}(E,1)/2! $ ≈ $ 13.822032139008053893342891034114435801 $

Modular invariants

Modular form 334620.2.a.l

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);
 

\( q - q^{5} - 2q^{7} + q^{11} - 2q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 10616832
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

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$(j)_{-}$
$2$ $3$ $IV^{*}$ Additive -1 2 8 0
$3$ $4$ $I_3^{*}$ Additive -1 2 9 3
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$13$ $4$ $I_0^{*}$ Additive 1 2 6 0

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 2.3.0.1
$3$ 3B 3.4.0.1

$p$-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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 334620l consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{3}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-39}) \) \(\Z/6\Z\) Not in database
$4$ 4.0.981552.2 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{3}, \sqrt{-13})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.2.197730000.3 \(\Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.138735983333376.115 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.8670998958336.28 \(\Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$18$ 18.0.296021858606759196583682402277234540734400000000.1 \(\Z/18\Z\) 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.