Properties

Label 2.2.29.1-121.1-a1
Base field \(\Q(\sqrt{29}) \)
Conductor \((11)\)
Conductor norm \( 121 \)
CM no
Base change yes: 9251.d1,11.a1
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{29}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 7 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 7)
 
gp: K = nfinit(a^2 - a - 7);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, -1, 1]);
 

Weierstrass equation

\(y^2+y=x^{3}-x^{2}-7820x-263580\)
sage: E = EllipticCurve(K, [0, -1, 1, -7820, -263580])
 
gp: E = ellinit([0, -1, 1, -7820, -263580],K)
 
magma: E := ChangeRing(EllipticCurve([0, -1, 1, -7820, -263580]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((11)\) = \( \left(11\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 121 \) = \( 121 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((11)\) = \( \left(11\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 121 \) = \( 121 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{52893159101157376}{11} \)
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 $\left(\frac{502612946675675601848897937446445135272531749}{2885470916238747904899554540514910677313600} : -\frac{1733883140214957773579429681208088271626538072218473638117813588159}{2450727681214765156982057884842646510916344935159382809427808000} a + \frac{1731432412533743008422447623323245625115621727283314255308385780159}{4901455362429530313964115769685293021832689870318765618855616000} : 1\right)$
Height \(103.183965783794\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 103.183965783794 \)
Period: \( 0.0644356903227915 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \(2.46927635593011\)
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\((j)_{-}\)
\( \left(11\right) \) \(121\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5 and 25.
Its isogeny class 121.1-a consists of curves linked by isogenies of degrees dividing 25.

Base change

This curve is the base change of elliptic curves 9251.d1, 11.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.