Properties

Label 3.3.49.1-91.1-a1
Base field \(\Q(\zeta_{7})^+\)
Conductor \((4 a + 1)\)
Conductor norm \( 91 \)
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

\(y^2+\left(a^{2}-2\right)xy=x^{3}+\left(a^{2}-3\right)x^{2}+\left(-1030a^{2}+1620a-484\right)x-21769a^{2}+41147a-14213\)
sage: E = EllipticCurve(K, [a^2 - 2, a^2 - 3, 0, -1030*a^2 + 1620*a - 484, -21769*a^2 + 41147*a - 14213])
 
gp: E = ellinit([a^2 - 2, a^2 - 3, 0, -1030*a^2 + 1620*a - 484, -21769*a^2 + 41147*a - 14213],K)
 
magma: E := ChangeRing(EllipticCurve([a^2 - 2, a^2 - 3, 0, -1030*a^2 + 1620*a - 484, -21769*a^2 + 41147*a - 14213]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4 a + 1)\) = \( \left(-a^{2} - a + 2\right) \cdot \left(-2 a^{2} + a + 2\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 91 \) = \( 7 \cdot 13 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((-168070 a^{2} + 67228 a + 218491)\) = \( \left(-a^{2} - a + 2\right)^{16} \cdot \left(-2 a^{2} + a + 2\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5616365266262569 \) = \( 7^{16} \cdot 13^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3586352161337298910516}{19882681} a^{2} - \frac{8058460190096647498093}{19882681} a + \frac{2876031146228402085760}{19882681} \)
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: $\left(-\frac{3}{4} a^{2} + \frac{95}{4} a - \frac{23}{4} : -\frac{69}{8} a^{2} + \frac{3}{8} a + \frac{23}{4} : 1\right)$
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: \( 0.284767587309665 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \(0.650897342422091\)
Analytic order of Ш: \( 16 \) (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\((j)_{-}\)
\( \left(-a^{2} - a + 2\right) \) \(7\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)
\( \left(-2 a^{2} + a + 2\right) \) \(13\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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, 4, 8 and 16.
Its isogeny class 91.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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