Properties

Label 3.3.49.1-41.3-a1
Base field \(\Q(\zeta_{7})^+\)
Conductor \((3 a^{2} - a - 3)\)
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: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)
 
gp: K = nfinit(a^3 - a^2 - 2*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
 

Weierstrass equation

\(y^2+xy=x^{3}+\left(a^{2}-a-3\right)x^{2}+\left(-91a^{2}+101a-68\right)x-646a^{2}+852a-304\)
sage: E = EllipticCurve(K, [1, a^2 - a - 3, 0, -91*a^2 + 101*a - 68, -646*a^2 + 852*a - 304])
 
gp: E = ellinit([1, a^2 - a - 3, 0, -91*a^2 + 101*a - 68, -646*a^2 + 852*a - 304],K)
 
magma: E := ChangeRing(EllipticCurve([1, a^2 - a - 3, 0, -91*a^2 + 101*a - 68, -646*a^2 + 852*a - 304]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3 a^{2} - a - 3)\) = \( \left(3 a^{2} - a - 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \( 41 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((-129569 a^{2} - 134116 a + 239850)\) = \( \left(3 a^{2} - a - 3\right)^{10} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 13422659310152401 \) = \( 41^{10} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{174656123697499408263335}{13422659310152401} a^{2} + \frac{182915726357803972950650}{13422659310152401} a - \frac{115913951592431810832436}{13422659310152401} \)
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: $\left(3 a^{2} - 11 a + \frac{11}{4} : -\frac{3}{2} a^{2} + \frac{11}{2} a - \frac{11}{8} : 1\right)$
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: \( 1.35782736618416 \)
Tamagawa product: \( 10 \)
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\((j)_{-}\)
\( \left(3 a^{2} - a - 3\right) \) \(41\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)

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.