# Properties

 Label 91.b2 Conductor 91 Discriminant -91 j-invariant $$-\frac{43614208}{91}$$ CM no Rank 1 Torsion Structure $$\Z/{3}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -7, 5]); // or
magma: E := EllipticCurve("91b1");
sage: E = EllipticCurve([0, 1, 1, -7, 5]) # or
sage: E = EllipticCurve("91b1")
gp: E = ellinit([0, 1, 1, -7, 5]) \\ or
gp: E = ellinit("91b1")

$$y^2 + y = x^{3} + x^{2} - 7 x + 5$$

## Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-1, 3\right)$$ $$\hat{h}(P)$$ ≈ 1.05924508641

## Torsion generators

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

$$\left(1, 0\right)$$

## Integral points

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

$$\left(-1, 3\right)$$, $$\left(1, 0\right)$$, $$\left(3, 4\right)$$

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: $$-\frac{43614208}{91}$$ = $$-1 \cdot 2^{15} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{-1}$$ 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: $$1.05924508641$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$6.03949153648$$ 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: $$3$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form91.2.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)

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

 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])

$$L'(E,1)$$ ≈ $$0.710811303826$$

## 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$$(j)_{-}$$
$$7$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$13$$ $$1$$ $$I_{1}$$ 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 $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B.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]

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

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss ordinary ordinary split ss split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 16,1 9 1 2 1,1 2 1 1 1 1 1 1 1 1 1 0,0 0 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 and 9.
Its isogeny class 91.b 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
3 3.3.169.1 $$\Z/9\Z$$ Not in database
3.1.364.1 $$\Z/6\Z$$ Not in database
6 6.0.1851523947.3 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.0.12057136.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.0.10955763.1 $$\Z/9\Z$$ 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.