Properties

Label 6.6.371293.1-79.3-c4
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{13})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+3a\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-435a^{5}-121a^{4}+1512a^{3}+155a^{2}-866a-246\right){x}-8307a^{5}-5881a^{4}+27378a^{3}+12730a^{2}-16010a-4422\)
sage: E = EllipticCurve([K([-2,-2,1,1,0,0]),K([2,-6,-1,5,0,-1]),K([0,3,-3,-4,1,1]),K([-246,-866,155,1512,-121,-435]),K([-4422,-16010,12730,27378,-5881,-8307])])
 
gp: E = ellinit([Polrev([-2,-2,1,1,0,0]),Polrev([2,-6,-1,5,0,-1]),Polrev([0,3,-3,-4,1,1]),Polrev([-246,-866,155,1512,-121,-435]),Polrev([-4422,-16010,12730,27378,-5881,-8307])], K);
 
magma: E := EllipticCurve([K![-2,-2,1,1,0,0],K![2,-6,-1,5,0,-1],K![0,3,-3,-4,1,1],K![-246,-866,155,1512,-121,-435],K![-4422,-16010,12730,27378,-5881,-8307]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-4a^3+3a-2)\) = \((a^5-4a^3+3a-2)\)
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: \((-1089a^5+183a^4+5613a^3-460a^2-6917a+671)\) = \((a^5-4a^3+3a-2)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -9468276082626847201 \) = \(-79^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{354107829458843827802411767457358588717502}{9468276082626847201} a^{5} + \frac{439473793889618077364787680142691941182070}{9468276082626847201} a^{4} + \frac{1664593722944885880969335611051937863705766}{9468276082626847201} a^{3} - \frac{1817720520510626376452406301038917963206945}{9468276082626847201} a^{2} - \frac{1686442982404405482780445086453016221746437}{9468276082626847201} a + \frac{1468879965454357964032378550143526285469230}{9468276082626847201} \)
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: $\left(\frac{25}{4} a^{5} - \frac{21}{2} a^{4} - 32 a^{3} + \frac{65}{2} a^{2} + \frac{109}{4} a - \frac{5}{4} : \frac{107}{8} a^{5} + \frac{17}{4} a^{4} - \frac{459}{8} a^{3} - \frac{19}{2} a^{2} + \frac{449}{8} a + \frac{77}{8} : 1\right)$
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: \( 2.9187900687961089692499549562740333975 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.49691 \)
Analytic order of Ш: \( 625 \) (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\((j)_{-}\)
\((a^5-4a^3+3a-2)\) \(79\) \(2\) \(I_{10}\) Non-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 79.3-c consists of curves linked by isogenies of degrees dividing 10.

Base change

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

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