Properties

Label 2.2.29.1-196.1-c2
Base field \(\Q(\sqrt{29}) \)
Conductor norm \( 196 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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

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

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

Weierstrass equation

\({y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-648a-1422\right){x}-4308a-9446\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([1,1]),K([-1422,-648]),K([-9446,-4308])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([1,1]),Polrev([-1422,-648]),Polrev([-9446,-4308])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![1,1],K![-1422,-648],K![-9446,-4308]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((14)\) = \((2)\cdot(-a)\cdot(a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 196 \) = \(4\cdot7\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((227360a+178752)\) = \((2)^{5}\cdot(-a)^{2}\cdot(a-1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 289254654976 \) = \(4^{5}\cdot7^{2}\cdot7^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{8271409004059059}{184473632} a + \frac{3772500342785869}{26353376} \)
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 $\left(-10 a - 22 : -2 a - 4 : 1\right)$
Height \(0.25559134011312380277357213194865090544\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{5}{4} a - \frac{11}{4} : \frac{3}{2} a + \frac{31}{8} : 1\right)$
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: \( 0.25559134011312380277357213194865090544 \)
Period: \( 6.1007552504833507237642494701074241589 \)
Tamagawa product: \( 16 \)  =  \(1\cdot2\cdot2^{3}\)
Torsion order: \(2\)
Leading coefficient: \( 2.3164382387813069695639714118836450681 \)
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\((j)_{-}\)
\((2)\) \(4\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((-a)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-1)\) \(7\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 196.1-c consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.