Properties

Label 11197.a1
Conductor 11197
Discriminant 11197
j-invariant \( \frac{20346417}{11197} \)
CM no
Rank 3
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -6, 0]) # or
 
sage: E = EllipticCurve("11197a1")
 
gp: E = ellinit([1, -1, 1, -6, 0]) \\ or
 
gp: E = ellinit("11197a1")
 
magma: E := EllipticCurve([1, -1, 1, -6, 0]); // or
 
magma: E := EllipticCurve("11197a1");
 

\( y^2 + x y + y = x^{3} - x^{2} - 6 x \)

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(-2, 1\right) \)\( \left(-1, 2\right) \)\( \left(0, 0\right) \)
\(\hat{h}(P)\) ≈  1.14086261134087061.0173402802513420.8657163846090297

Integral points

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

\( \left(-2, 1\right) \), \( \left(-2, 0\right) \), \( \left(-1, 2\right) \), \( \left(-1, -2\right) \), \( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(3, 0\right) \), \( \left(3, -4\right) \), \( \left(4, 3\right) \), \( \left(4, -8\right) \), \( \left(6, 9\right) \), \( \left(6, -16\right) \), \( \left(8, 16\right) \), \( \left(8, -25\right) \), \( \left(10, 24\right) \), \( \left(10, -35\right) \), \( \left(31, 154\right) \), \( \left(31, -186\right) \), \( \left(50, 325\right) \), \( \left(50, -376\right) \), \( \left(95, 874\right) \), \( \left(95, -970\right) \), \( \left(259, 4032\right) \), \( \left(259, -4292\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 11197 \)  =  \(11197\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(11197 \)  =  \(11197 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{20346417}{11197} \)  =  \(3^{3} \cdot 7^{3} \cdot 13^{3} \cdot 11197^{-1}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(3\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.868421953694\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(3.30667107387\)
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: \( 1 \)  = \( 1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 11197.2.a.a

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);
 

\( q - q^{2} - 3q^{3} - q^{4} - 2q^{5} + 3q^{6} - 5q^{7} + 3q^{8} + 6q^{9} + 2q^{10} - 4q^{11} + 3q^{12} - 7q^{13} + 5q^{14} + 6q^{15} - q^{16} - 2q^{17} - 6q^{18} - 5q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2080
\( \Gamma_0(N) \)-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[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

\( L^{(3)}(E,1)/3! \) ≈ \( 2.87158575419 \)

Local data

This elliptic curve is semistable.

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\((j)_{-}\)
\(11197\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

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 \( p \) .

$p$-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 11197
Reduction type ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) 3 3,3 3 9 3 3 3 3 3 3,3 3 3 3 3 3 ?
$\mu$-invariant(s) 0 0,0 0 0 0 0 0 0 0 0,0 0 0 0 0 0 ?

An entry ? indicates that the invariants have not yet been computed.

Isogenies

This curve has no rational isogenies. Its isogeny class 11197.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 3.3.11197.1 \(\Z/2\Z\) Not in database
6 6.6.1403799342373.1 \(\Z/2\Z \times \Z/2\Z\) 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.