Properties

Label 3.3.49.1-181.3-a1
Base field \(\Q(\zeta_{7})^+\)
Conductor norm \( 181 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{7})^+\)

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 2 x + 1 \); class number \(1\).

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

Weierstrass equation

\({y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(274a^{2}-170a-634\right){x}+2618a^{2}-1512a-5956\)
sage: E = EllipticCurve([K([-1,0,1]),K([-2,0,1]),K([-1,1,1]),K([-634,-170,274]),K([-5956,-1512,2618])])
 
gp: E = ellinit([Polrev([-1,0,1]),Polrev([-2,0,1]),Polrev([-1,1,1]),Polrev([-634,-170,274]),Polrev([-5956,-1512,2618])], K);
 
magma: E := EllipticCurve([K![-1,0,1],K![-2,0,1],K![-1,1,1],K![-634,-170,274],K![-5956,-1512,2618]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-5a-2)\) = \((a^2-5a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 181 \) = \(181\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1042a^2-4798a-3419)\) = \((a^2-5a-2)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -194264244901 \) = \(-181^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{44023832204770836564641}{194264244901} a^{2} - \frac{24431384735829497330761}{194264244901} a - \frac{98920654043308873836255}{194264244901} \)
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: \(0\)
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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.1678486976859120383081642361751133729 \)
Tamagawa product: \( 5 \)
Torsion order: \(1\)
Leading coefficient: \( 0.83417764120422288450583159726793812352 \)
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\((j)_{-}\)
\((a^2-5a-2)\) \(181\) \(5\) \(I_{5}\) 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
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 181.3-a consists of curves linked by isogenies of degree 5.

Base change

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

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