Properties

Label 6.6.371293.1-79.4-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

\({y}^2+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+3a-2\right){y}={x}^{3}+\left(-a^{5}-a^{4}+6a^{3}+4a^{2}-9a-1\right){x}^{2}+\left(-947a^{5}+2047a^{4}+2349a^{3}-6681a^{2}+2027a+739\right){x}-37043a^{5}+78539a^{4}+95061a^{3}-254944a^{2}+69625a+31719\)
sage: E = EllipticCurve([K([1,-2,-3,1,1,0]),K([-1,-9,4,6,-1,-1]),K([-2,3,1,-4,0,1]),K([739,2027,-6681,2349,2047,-947]),K([31719,69625,-254944,95061,78539,-37043])])
 
gp: E = ellinit([Polrev([1,-2,-3,1,1,0]),Polrev([-1,-9,4,6,-1,-1]),Polrev([-2,3,1,-4,0,1]),Polrev([739,2027,-6681,2349,2047,-947]),Polrev([31719,69625,-254944,95061,78539,-37043])], K);
 
magma: E := EllipticCurve([K![1,-2,-3,1,1,0],K![-1,-9,4,6,-1,-1],K![-2,3,1,-4,0,1],K![739,2027,-6681,2349,2047,-947],K![31719,69625,-254944,95061,78539,-37043]]);
 

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: \((-938a^5-466a^4+4599a^3+1277a^2-5125a-377)\) = \((-a^4+a^3+3a^2-3a-2)^{10}\)
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: \( -\frac{59825344952154066993255580468150198478665}{9468276082626847201} a^{5} + \frac{165770769301487325035978806703005278360409}{9468276082626847201} a^{4} + \frac{70395843676331013626156886098889571019}{119851595982618319} a^{3} - \frac{249149902794951405348247878886512326245469}{9468276082626847201} a^{2} + \frac{82270495728478075345113362065223016794412}{9468276082626847201} a + \frac{33782222503946602790170526944985186954909}{9468276082626847201} \)
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: $\left(-12 a^{5} + 14 a^{4} + \frac{165}{4} a^{3} - \frac{99}{2} a^{2} - 7 a + 16 : 4 a^{5} + \frac{7}{2} a^{4} - \frac{127}{8} a^{3} - \frac{53}{8} a^{2} + \frac{143}{8} a - \frac{7}{2} : 1\right)$
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: \( 2.9187900687961089692499549562740333975 \)
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\((j)_{-}\)
\((-a^4+a^3+3a^2-3a-2)\) \(79\) \(2\) \(I_{10}\) 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 79.4-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.