Properties

Label 2.2.29.1-121.1-a2
Base field \(\Q(\sqrt{29}) \)
Conductor \((11)\)
Conductor norm \( 121 \)
CM no
Base change yes: 11.a2,9251.d2
Q-curve yes
Torsion order \( 5 \)
Rank \( 1 \)

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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

\(y^2+y=x^{3}-x^{2}-10x-20\)
sage: E = EllipticCurve(K, [0, -1, 1, -10, -20])
 
gp: E = ellinit([0, -1, 1, -10, -20],K)
 
magma: E := ChangeRing(EllipticCurve([0, -1, 1, -10, -20]),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: \((161051)\) = \( \left(11\right)^{5} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 25937424601 \) = \( 121^{5} \)
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 $\left(\frac{907428789}{5569600} : -\frac{5059406780519}{6572128000} a + \frac{5052834652519}{13144256000} : 1\right)$
Height \(20.6367931567589\)
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: \( 20.6367931567589 \)
Period: \( 1.61089225806979 \)
Tamagawa product: \( 5 \)
Torsion order: \(5\)
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\((j)_{-}\)
\( \left(11\right) \) \(121\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

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 121.1-a consists of curves linked by isogenies of degrees dividing 25.

Base change

This curve is the base change of elliptic curves 11.a2, 9251.d2, defined over \(\Q\), so it is also a \(\Q\)-curve.