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Properties

Label 6.6.485125.1-79.1-a1
Base field 6.6.485125.1
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
 
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-5a^{3}+5a+1\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+3a^{2}+2a\right){y}={x}^{3}+\left(3a^{5}-3a^{4}-14a^{3}+10a^{2}+13a-2\right){x}^{2}+\left(14a^{5}-9a^{4}-62a^{3}+35a^{2}+54a-11\right){x}+23a^{5}-12a^{4}-97a^{3}+48a^{2}+79a-22\)
sage: E = EllipticCurve([K([1,5,0,-5,0,1]),K([-2,13,10,-14,-3,3]),K([0,2,3,-4,-1,1]),K([-11,54,35,-62,-9,14]),K([-22,79,48,-97,-12,23])])
 
gp: E = ellinit([Polrev([1,5,0,-5,0,1]),Polrev([-2,13,10,-14,-3,3]),Polrev([0,2,3,-4,-1,1]),Polrev([-11,54,35,-62,-9,14]),Polrev([-22,79,48,-97,-12,23])], K);
 
magma: E := EllipticCurve([K![1,5,0,-5,0,1],K![-2,13,10,-14,-3,3],K![0,2,3,-4,-1,1],K![-11,54,35,-62,-9,14],K![-22,79,48,-97,-12,23]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4+a^3-3a^2-3a+2)\) = \((a^4+a^3-3a^2-3a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 79 \) = \(79\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^4+2a^3+3a^2-5a)\) = \((a^4+a^3-3a^2-3a+2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 79 \) = \(79\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{34707474083885637}{79} a^{5} - \frac{4278503312080601}{79} a^{4} + \frac{129745464680528828}{79} a^{3} - \frac{2174717524050824}{79} a^{2} - \frac{74032467724509348}{79} a + \frac{16346211679544252}{79} \)
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: \( 1151.1764914148381682189438908368566239 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.65278 \)
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\((j)_{-}\)
\((a^4+a^3-3a^2-3a+2)\) \(79\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 79.1-a consists of this curve only.

Base change

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

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