Properties

Label 91b2
Conductor 91
Discriminant -753571
j-invariant \( \frac{224755712}{753571} \)
CM no
Rank 1
Torsion Structure \(\Z/{3}\Z\)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, 13, 42]); // or
magma: E := EllipticCurve("91b2");
sage: E = EllipticCurve([0, 1, 1, 13, 42]) # or
sage: E = EllipticCurve("91b2")
gp: E = ellinit([0, 1, 1, 13, 42]) \\ or
gp: E = ellinit("91b2")

\( y^2 + y = x^{3} + x^{2} + 13 x + 42 \)

Mordell-Weil group structure

\(\Z\times \Z/{3}\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-2, 3\right) \)
\(\hat{h}(P)\) ≈  0.35308169547

Torsion generators

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

\( \left(0, 6\right) \)

Integral points

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

\( \left(-2, 3\right) \), \( \left(0, 6\right) \), \( \left(12, 45\right) \), \( \left(26, 136\right) \), \( \left(130, 1488\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 91 \)  =  \(7 \cdot 13\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-753571 \)  =  \(-1 \cdot 7^{3} \cdot 13^{3} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{224755712}{753571} \)  =  \(2^{15} \cdot 7^{-3} \cdot 13^{-3} \cdot 19^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(1\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(0.35308169547\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(2.01316384549\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
Tamagawa product: \( 9 \)  = \( 3\cdot3 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(3\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 91.2.a.b

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

\( q - 2q^{3} - 2q^{4} - 3q^{5} + q^{7} + q^{9} + 4q^{12} + q^{13} + 6q^{15} + 4q^{16} - 6q^{17} - 7q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 12
\( \Gamma_0(N) \)-optimal: no
Manin constant: not computed

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

\( L'(E,1) \) ≈ \( 0.710811303826 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(7\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(13\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

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.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 ss ordinary ordinary split ss split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 16,1 9 1 2 1,1 2 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) 0,0 1 0 0 0,0 0 0 0 0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 91b consists of 3 curves linked by isogenies of degrees dividing 9.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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
2 \(\Q(\sqrt{-3}) \) \(\Z/3\Z \times \Z/3\Z\) 2.0.3.1-8281.5-a3
3 3.1.364.1 \(\Z/6\Z\) Not in database
6 6.0.12057136.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
6.0.3577392.1 \(\Z/3\Z \times \Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.