# Properties

 Label 11.a2 Conductor $11$ Discriminant $-161051$ j-invariant $$-\frac{122023936}{161051}$$ CM no Rank $0$ Torsion structure $$\Z/{5}\Z$$

# Related objects

Show commands: 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])

gp: E = ellinit([0, -1, 1, -10, -20])

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

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

## 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)[1]  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: $\mathrm{SU}(2)$ Faltings height: $-0.30800984111840306468901426146\dots$ Stable Faltings height: $-0.30800984111840306468901426146\dots$

## 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: $1.2692093042795534216887946168\dots$ 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: $5$  = $5$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $5$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) 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); Special value: $L(E,1)$ ≈ $0.25384186085591068433775892335043887465$

## Modular invariants

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);

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

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

## Local data

This elliptic curve is semistable. There is only one prime 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$(j)_{-}$
$11$ $5$ $I_{5}$ Split multiplicative -1 1 5 5

## Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$5$ 5Cs.1.1 5.120.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 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 $8$ 8.2.32019867.1 $$\Z/15\Z$$ Not in database $12$ 12.2.20433779818496.3 $$\Z/20\Z$$ Not in database $12$ 12.0.7320500000000.2 $$\Z/5\Z \times \Z/10\Z$$ Not in database

We only show fields where the torsion growth is primitive.