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Properties

Label 147.b2
Conductor $147$
Discriminant $-147$
j-invariant \( -\frac{28672}{3} \)
CM no
Rank $0$
Torsion structure trivial

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This isogeny class and its quadratic twist by $\Q(\sqrt{-3})$ are the ones of minimal conductor with a $13$-isogeny.

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -2, -1]) # or
 
sage: E = EllipticCurve("147.b2")
 
gp: E = ellinit([0, -1, 1, -2, -1]) \\ or
 
gp: E = ellinit("147.b2")
 
magma: E := EllipticCurve([0, -1, 1, -2, -1]); // or
 
magma: E := EllipticCurve("147.b2");
 

\(y^2+y=x^3-x^2-2x-1\)

Mordell-Weil group structure

trivial

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 147 \)  =  \(3 \cdot 7^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-147 \)  =  \(-1 \cdot 3 \cdot 7^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{28672}{3} \)  =  \(-1 \cdot 2^{12} \cdot 3^{-1} \cdot 7\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

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: \(1.9200537453490419433285021703\)
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: \( 1 \)  = \( 1\cdot1 \)
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   147.2.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);
 

\( q + 2q^{2} - q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + q^{9} + 4q^{10} - 2q^{11} - 2q^{12} - q^{13} - 2q^{15} - 4q^{16} + 2q^{18} - q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 6
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 1.9200537453490419433285021702840752175 \)

Local data

This elliptic curve is semistable. There are 2 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\((j)_{-}\)
\(3\) \(1\) \(I_{1}\) Non-split multiplicative 1 1 1 1
\(7\) \(1\) \(II\) Additive -1 2 2 0

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
\(13\) B.4.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 nonsplit ordinary add ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 0,1 0 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 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 \(d=\) 13.
Its isogeny class 147.b consists of 2 curves linked by isogenies of degree 13.

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
$3$ 3.1.588.1 \(\Z/2\Z\) Not in database
$6$ 6.0.1037232.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ \(\Q(\zeta_{7})\) \(\Z/13\Z\) Not in database
$8$ 8.2.20841167403.1 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$18$ 18.0.14176142707705638912.1 \(\Z/26\Z\) 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.