# Properties

 Label 11a3 Conductor $11$ Discriminant $-11$ j-invariant $$-\frac{4096}{11}$$ CM no Rank $0$ Torsion structure $$\Z/{5}\Z$$

# Learn more

Show commands: Magma / Oscar / PariGP / SageMath

The first elliptic curve in nature. This is a model for the modular curve $$X_1(11)$$.

## Simplified equation

 $$y^2+y=x^3-x^2$$ y^2+y=x^3-x^2 (homogenize, simplify) $$y^2z+yz^2=x^3-x^2z$$ y^2z+yz^2=x^3-x^2z (dehomogenize, simplify) $$y^2=x^3-432x+8208$$ y^2=x^3-432x+8208 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, -1, 1, 0, 0])

gp: E = ellinit([0, -1, 1, 0, 0])

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

oscar: E = EllipticCurve([0, -1, 1, 0, 0])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

## Torsion generators

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

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

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

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$11$$ = $11$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-11$ = $-1 \cdot 11$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{4096}{11}$$ = $-1 \cdot 2^{12} \cdot 11^{-1}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-1.1127287973354532519893939281\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.1127287973354532519893939281\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.8254556483942886\dots$ Szpiro ratio: $4.190241815676422\dots$

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $6.3460465213977671084439730838\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $1$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $5$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ ( exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $0.25384186085591068433775892335$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 0.253841861 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 6.346047 \cdot 1.000000 \cdot 1}{5^2} \approx 0.253841861$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

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

Modular degree: 5
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 5
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is semistable. There is only one prime of bad reduction:

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

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

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

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$5$ 5B.1.1 25.120.0.1

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[316, 545, 377, 287], [38, 41, 191, 539], [1, 0, 50, 1], [1, 50, 0, 1], [501, 50, 500, 51], [516, 5, 35, 186]]

GL(2,Integers(550)).subgroup(gens)

Gens := [[316, 545, 377, 287], [38, 41, 191, 539], [1, 0, 50, 1], [1, 50, 0, 1], [501, 50, 500, 51], [516, 5, 35, 186]];

sub<GL(2,Integers(550))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$550 = 2 \cdot 5^{2} \cdot 11$$, index $1200$, genus $37$, and generators

$\left(\begin{array}{rr} 316 & 545 \\ 377 & 287 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 191 & 539 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 501 & 50 \\ 500 & 51 \end{array}\right),\left(\begin{array}{rr} 516 & 5 \\ 35 & 186 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[550])$ is a degree-$19800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/550\Z)$.

## Isogenies

gp: ellisomat(E)

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

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $5$ $$\Q(\zeta_{11})^+$$ $$\Z/25\Z$$ 5.5.14641.1-11.1-a3 $6$ 6.0.21296.1 $$\Z/2\Z \oplus \Z/10\Z$$ 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 $15$ 15.5.35351257235385344.1 $$\Z/50\Z$$ Not in database $20$ 20.0.1402274470934209014892578125.2 $$\Z/5\Z \oplus \Z/5\Z$$ Not in database $20$ 20.0.95777233176300048828125.1 $$\Z/25\Z$$ Not in database

We only show fields where the torsion growth is primitive.

## Iwasawa invariants

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

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

## $p$-adic regulators

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