Properties

Label 3.3.49.1-41.3-a2
Base field \(\Q(\zeta_{7})^+\)
Conductor \((3a^2-a-3)\)
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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(Pol(Vecrev([1, -2, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-71a^{2}+146a-43\right){x}-456a^{2}+1027a-381\)
sage: E = EllipticCurve([K([1,0,0]),K([-3,-1,1]),K([0,0,0]),K([-43,146,-71]),K([-381,1027,-456])])
 
gp: E = ellinit([Pol(Vecrev([1,0,0])),Pol(Vecrev([-3,-1,1])),Pol(Vecrev([0,0,0])),Pol(Vecrev([-43,146,-71])),Pol(Vecrev([-381,1027,-456]))], K);
 
magma: E := EllipticCurve([K![1,0,0],K![-3,-1,1],K![0,0,0],K![-43,146,-71],K![-381,1027,-456]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3a^2-a-3)\) = \((3a^2-a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \(41\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((113a^2-400a-372)\) = \((3a^2-a-3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 115856201 \) = \(41^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{653981503916958524755}{115856201} a^{2} - \frac{1469483101129546552831}{115856201} a + \frac{524452446825637320235}{115856201} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-2 a^{2} + 6 a : a^{2} - 3 a : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2.71565473236833 \)
Tamagawa product: \( 5 \)
Torsion order: \(2\)
Leading coefficient: \( 0.484938345065773 \)
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\((j)_{-}\)
\((3a^2-a-3)\) \(41\) \(5\) \(I_{5}\) 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
\(2\) 2B
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 41.3-a consists of curves linked by isogenies of degrees dividing 10.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.