Properties

Label 3330ba2
Conductor $3330$
Discriminant $-4.143\times 10^{13}$
j-invariant \( -\frac{39390416456458249}{56832000000} \)
CM no
Rank $1$
Torsion structure \(\Z/{3}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -63797, 6225869])
 
gp: E = ellinit([1, -1, 1, -63797, 6225869])
 
magma: E := EllipticCurve([1, -1, 1, -63797, 6225869]);
 

\(y^2+xy+y=x^3-x^2-63797x+6225869\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{3}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(147, 16\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.19332285560794357927682766401$

Torsion generators

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

\( \left(127, 336\right) \)  Toggle raw display

Integral points

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

\( \left(-273, 1936\right) \), \( \left(-273, -1664\right) \), \( \left(-33, 2896\right) \), \( \left(-33, -2864\right) \), \( \left(57, 1636\right) \), \( \left(57, -1694\right) \), \( \left(103, 808\right) \), \( \left(103, -912\right) \), \( \left(127, 336\right) \), \( \left(127, -464\right) \), \( \left(147, 16\right) \), \( \left(147, -164\right) \), \( \left(159, 208\right) \), \( \left(159, -368\right) \), \( \left(177, 586\right) \), \( \left(177, -764\right) \), \( \left(327, 4336\right) \), \( \left(327, -4664\right) \), \( \left(687, 16576\right) \), \( \left(687, -17264\right) \), \( \left(2527, 125136\right) \), \( \left(2527, -127664\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 3330 \)  =  $2 \cdot 3^{2} \cdot 5 \cdot 37$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-41430528000000 $  =  $-1 \cdot 2^{15} \cdot 3^{7} \cdot 5^{6} \cdot 37 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{39390416456458249}{56832000000} \)  =  $-1 \cdot 2^{-15} \cdot 3^{-1} \cdot 5^{-6} \cdot 7^{3} \cdot 13^{3} \cdot 37^{-1} \cdot 3739^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.5153225683768351292118076122\dots$
Stable Faltings height: $0.96601642404278028351418499374\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.19332285560794357927682766401\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.64299126178331434284211379163\dots$
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: $ 360 $  = $ ( 3 \cdot 5 )\cdot2^{2}\cdot( 2 \cdot 3 )\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $3$
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) $ ≈ $ 4.9721962743562051701183226193195567392 $

Modular invariants

Modular form   3330.2.a.u

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);
 

\( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - 3q^{11} - 7q^{13} - q^{14} + q^{16} + 3q^{17} - q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 17280
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $15$ $I_{15}$ Split multiplicative -1 1 15 15
$3$ $4$ $I_1^{*}$ Additive -1 2 7 1
$5$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$37$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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
$3$ 3B.1.1 3.8.0.1

$p$-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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split ss ordinary ordinary
$\lambda$-invariant(s) 4 - 2 1 1 1 1 1 1 1 1 2 1,1 1 1
$\mu$-invariant(s) 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 3330ba 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.888.1 \(\Z/6\Z\) Not in database
$6$ 6.0.700227072.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.332001998667.2 \(\Z/3\Z \times \Z/3\Z\) Not in database
$9$ 9.3.26892161892027000000.5 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$18$ 18.0.13133045975992791649217115575980427141971968.1 \(\Z/3\Z \times \Z/6\Z\) 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.