Properties

Label 6.6.371293.1-79.3-c3
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

\({y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+4a+4\right){x}^{2}+\left(11a^{5}-45a^{4}-15a^{3}+102a^{2}+9a-54\right){x}+59a^{5}-254a^{4}-57a^{3}+539a^{2}+2a-247\)
sage: E = EllipticCurve([K([0,5,-3,-5,1,1]),K([4,4,-5,-1,1,0]),K([3,2,-4,-4,1,1]),K([-54,9,102,-15,-45,11]),K([-247,2,539,-57,-254,59])])
 
gp: E = ellinit([Polrev([0,5,-3,-5,1,1]),Polrev([4,4,-5,-1,1,0]),Polrev([3,2,-4,-4,1,1]),Polrev([-54,9,102,-15,-45,11]),Polrev([-247,2,539,-57,-254,59])], K);
 
magma: E := EllipticCurve([K![0,5,-3,-5,1,1],K![4,4,-5,-1,1,0],K![3,2,-4,-4,1,1],K![-54,9,102,-15,-45,11],K![-247,2,539,-57,-254,59]]);
 

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: \((-13a^5+32a^4+48a^3-107a^2-50a+35)\) = \((a^5-4a^3+3a-2)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -3077056399 \) = \(-79^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{15509334442908689571428}{3077056399} a^{5} + \frac{44198347225679886465327}{3077056399} a^{4} - \frac{1469794761354986220538}{3077056399} a^{3} - \frac{65596401146446137115029}{3077056399} a^{2} + \frac{25462955612740497650596}{3077056399} a + \frac{8445950409748499043000}{3077056399} \)
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: $\left(-2 a^{5} + 2 a^{4} + 12 a^{3} - 8 a^{2} - 13 a + 5 : -4 a^{5} + 14 a^{3} - 8 a + 2 : 1\right)$
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: \( 5.8375801375922179384999099125480667949 \)
Tamagawa product: \( 1 \)
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\((j)_{-}\)
\((a^5-4a^3+3a-2)\) \(79\) \(1\) \(I_{5}\) Non-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 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.