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Properties

Label 361.b2
Conductor $361$
Discriminant $-322687697779$
j-invariant \( -\frac{89915392}{6859} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -3369, 81208])
 
gp: E = ellinit([0, -1, 1, -3369, 81208])
 
magma: E := EllipticCurve([0, -1, 1, -3369, 81208]);
 

\(y^2+y=x^3-x^2-3369x+81208\)  Toggle raw display

Mordell-Weil group structure

trivial

Integral points

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

\(\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 361 \)  =  \(19^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-322687697779 \)  =  \(-1 \cdot 19^{9} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{89915392}{6859} \)  =  \(-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(0.95635250224048318782698598107\dots\)
Stable Faltings height: \(-0.51586698734273704217752773487\dots\)

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: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.94681992979742269053643643614\dots\)
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: \( 2 \)  = \( 2 \)
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   361.2.a.b

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);
 

\( q + 2q^{3} - 2q^{4} + 3q^{5} - q^{7} + q^{9} + 3q^{11} - 4q^{12} + 4q^{13} + 6q^{15} + 4q^{16} - 3q^{17} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 360
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 1.8936398595948453810728728722731192907 \)

Local data

This elliptic curve is not semistable. There is only one prime 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\((j)_{-}\)
\(19\) \(2\) \(I_3^{*}\) Additive -1 2 9 3

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
\(3\) Cs

$p$-adic data

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ordinary ordinary ordinary ordinary ordinary ordinary add ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) ? 0 0 0 0 0 0 - 0,0 0 0 0 0 0 0
$\mu$-invariant(s) ? 0 0 0 0 0 0 - 0,0 0 0 0 0 0 0

An entry ? indicates that the invariants have not yet been computed.

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 361.b consists of 2 curves linked by isogenies of degrees dividing 9.

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
$2$ \(\Q(\sqrt{57}) \) \(\Z/3\Z\) 2.2.57.1-19.1-b2
$2$ \(\Q(\sqrt{-19}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.76.1 \(\Z/2\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-19})\) \(\Z/3\Z \times \Z/3\Z\) Not in database
$6$ 6.0.109744.2 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.2.2963088.1 \(\Z/6\Z\) Not in database
$12$ 12.2.937292452593664.2 \(\Z/4\Z\) Not in database
$12$ 12.0.8779890495744.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$18$ 18.6.115765164098281427122348305637113.3 \(\Z/9\Z\) Not in database
$18$ 18.0.11067041604778727959971.1 \(\Z/9\Z\) Not in database

We only show fields where the torsion growth is primitive.