Show commands for: Magma / Pari/GP / SageMath

This is a model for the modular curve $$X_0(11)$$.

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -10, -20]) # or

sage: E = EllipticCurve("11a1")

gp: E = ellinit([0, -1, 1, -10, -20]) \\ or

gp: E = ellinit("11a1")

magma: E := EllipticCurve([0, -1, 1, -10, -20]); // or

magma: E := EllipticCurve("11a1");

$$y^2 + y = x^{3} - x^{2} - 10 x - 20$$

## Mordell-Weil group structure

$$\Z/{5}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(5, 5\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(5, 5\right)$$, $$\left(5, -6\right)$$, $$\left(16, 60\right)$$, $$\left(16, -61\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$11$$ = $$11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-161051$$ = $$-1 \cdot 11^{5}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{122023936}{161051}$$ = $$-1 \cdot 2^{12} \cdot 11^{-5} \cdot 31^{3}$$ 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); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$1.26920930428$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$5$$  = $$5$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$5$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} - 2q^{7} - 2q^{9} - 2q^{10} + q^{11} - 2q^{12} + 4q^{13} + 4q^{14} - q^{15} - 4q^{16} - 2q^{17} + 4q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1 $$\Gamma_0(N)$$-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/factorial(ar)

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L(E,1)$$ ≈ $$0.253841860856$$

## Local data

This elliptic curve is semistable.

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$11$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5

## 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
$$5$$ 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]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 11 ss ordinary ordinary split 0,1 0 0 1 0,0 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 11.a consists of 3 curves linked by isogenies of degrees dividing 25.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.44.1 $$\Z/10\Z$$ Not in database
$4$ $$\Q(\zeta_{5})$$ $$\Z/5\Z \times \Z/5\Z$$ Not in database
$6$ 6.0.21296.1 $$\Z/2\Z \times \Z/10\Z$$ 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.