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Properties

Label 966.d1
Conductor $966$
Discriminant $-85501108224$
j-invariant \( -\frac{14943832855786297}{85501108224} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -5131, -144323])
 
gp: E = ellinit([1, 1, 0, -5131, -144323])
 
magma: E := EllipticCurve([1, 1, 0, -5131, -144323]);
 

\(y^2+xy=x^3+x^2-5131x-144323\)  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: \( 966 \)  =  \(2 \cdot 3 \cdot 7 \cdot 23\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-85501108224 \)  =  \(-1 \cdot 2^{13} \cdot 3^{3} \cdot 7^{5} \cdot 23 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{14943832855786297}{85501108224} \)  =  \(-1 \cdot 2^{-13} \cdot 3^{-3} \cdot 7^{-5} \cdot 17^{3} \cdot 23^{-1} \cdot 14489^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(0.93940292624117495481286801118\dots\)
Stable Faltings height: \(0.93940292624117495481286801118\dots\)

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: \(0.28194180971556692829762487041\dots\)
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: \( 5 \)  = \( 1\cdot1\cdot5\cdot1 \)
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   966.2.a.d

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);
 

\( q - q^{2} - q^{3} + q^{4} + 3q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 3q^{10} + 4q^{11} - q^{12} + 3q^{13} - q^{14} - 3q^{15} + q^{16} - q^{18} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1560
\( \Gamma_0(N) \)-optimal: yes
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);
 

\( L(E,1) \) ≈ \( 1.4097090485778346414881243520474933950 \)

Local data

This elliptic curve is 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\((j)_{-}\)
\(2\) \(1\) \(I_{13}\) Non-split multiplicative 1 1 13 13
\(3\) \(1\) \(I_{3}\) Non-split multiplicative 1 1 3 3
\(7\) \(5\) \(I_{5}\) Split multiplicative -1 1 5 5
\(23\) \(1\) \(I_{1}\) Non-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 \( p \) .

$p$-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 nonsplit nonsplit ordinary split ordinary ordinary ss ss nonsplit ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 1 0 6 1 0 0 0,0 0,0 0 2 0 0 0 0 0
$\mu$-invariant(s) 0 0 0 0 0 0 0,0 0,0 0 0 0 0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 966.d consists of this curve only.

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
$3$ 3.1.3864.1 \(\Z/2\Z\) Not in database
$6$ 6.0.57691436544.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ 8.2.23511063249072.7 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) 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.