Properties

Label 2.2.209.1-11.1-a2
Base field \(\Q(\sqrt{209}) \)
Conductor norm \( 11 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 5 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{209}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 52 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-52, -1, 1]))
 
gp: K = nfinit(Polrev([-52, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20\)
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([-10,0]),K([-20,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-10,0]),Polrev([-20,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-10,0],K![-20,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((70a+471)\) = \((70a+471)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 11 \) = \(11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-161051)\) = \((70a+471)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 25937424601 \) = \(11^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{122023936}{161051} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(\frac{111231}{1100} : \frac{8464183}{60500} a - \frac{8524683}{121000} : 1\right)$
Height \(11.636512033654783699795904425058588313\)
Torsion structure: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(5 : -6 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 11.636512033654783699795904425058588313 \)
Period: \( 1.6108922580697880236230571558378844726 \)
Tamagawa product: \( 10 \)
Torsion order: \(5\)
Leading coefficient: \( 1.0373042593952838872347805656910110184 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((70a+471)\) \(11\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(5\) 5Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 11.1-a consists of curves linked by isogenies of degrees dividing 25.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 11.a2
\(\Q\) 43681.a2