Properties

Label 91.a1
Conductor 91
Discriminant -91
j-invariant \( \frac{110592}{91} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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This is a model for the quotient of the modular curve $X_0(91)$ by its group $\langle w_7, w_{13} \rangle$ of Atkin-Lehner involutions.

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, 1, 0]); // or
magma: E := EllipticCurve("91a1");
sage: E = EllipticCurve([0, 0, 1, 1, 0]) # or
sage: E = EllipticCurve("91a1")
gp: E = ellinit([0, 0, 1, 1, 0]) \\ or
gp: E = ellinit("91a1")

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

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(0, 0\right) \)
\(\hat{h}(P)\) ≈  0.142392150669

Integral points

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

\( \left(0, 0\right) \), \( \left(1, 1\right) \), \( \left(3, 5\right) \)

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: \(-91 \)  =  \(-1 \cdot 7 \cdot 13 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{110592}{91} \)  =  \(2^{12} \cdot 3^{3} \cdot 7^{-1} \cdot 13^{-1}\)
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: \(0.142392150669\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(3.89726093705\)
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: \( 1 \)  = \( 1\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 91.2.a.a

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)

\( q - 2q^{2} + 2q^{4} - 3q^{5} - q^{7} - 3q^{9} + 6q^{10} - 6q^{11} - q^{13} + 2q^{14} - 4q^{16} + 4q^{17} + 6q^{18} + 5q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 4
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

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])

\( L'(E,1) \) ≈ \( 0.554939366544 \)

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\((j)_{-}\)
\(7\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(13\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

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 \( p \) .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ss ordinary nonsplit ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 16,1 1,7 1 1 1 1 3 1 1 1 1 1 1 1 1
$\mu$-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 91.a consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.364.1 \(\Z/2\Z\) Not in database
6 6.0.12057136.1 \(\Z/2\Z \times \Z/2\Z\) 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.