Properties

Label 6.6.371293.1-79.3-d5
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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

\({y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-4a-3\right){x}^{2}+\left(4a^{5}-a^{4}-15a^{3}-16a^{2}+26a-2\right){x}-30a^{5}+79a^{4}+46a^{3}-216a^{2}+53a+38\)
sage: E = EllipticCurve([K([1,-2,0,1,0,0]),K([-3,-4,4,1,-1,0]),K([1,-3,-3,1,1,0]),K([-2,26,-16,-15,-1,4]),K([38,53,-216,46,79,-30])])
 
gp: E = ellinit([Polrev([1,-2,0,1,0,0]),Polrev([-3,-4,4,1,-1,0]),Polrev([1,-3,-3,1,1,0]),Polrev([-2,26,-16,-15,-1,4]),Polrev([38,53,-216,46,79,-30])], K);
 
magma: E := EllipticCurve([K![1,-2,0,1,0,0],K![-3,-4,4,1,-1,0],K![1,-3,-3,1,1,0],K![-2,26,-16,-15,-1,4],K![38,53,-216,46,79,-30]]);
 

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: \((-4a^4+a^3+17a^2-2a-9)\) = \((a^5-4a^3+3a-2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6241 \) = \(79^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2775957273063928}{6241} a^{5} + \frac{2158369015824389}{6241} a^{4} + \frac{12719957740377180}{6241} a^{3} - \frac{8693763912465636}{6241} a^{2} - \frac{11508811251959304}{6241} a + \frac{7949534780236512}{6241} \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(2 a^{5} - 2 a^{4} - 9 a^{3} + 7 a^{2} + 6 a - 2 : -a^{5} + 2 a^{4} + 3 a^{3} - 6 a^{2} + a + 2 : 1\right)$ $\left(-\frac{1}{4} a^{5} + \frac{3}{4} a^{4} + \frac{3}{2} a^{3} - \frac{11}{2} a^{2} + \frac{1}{4} a + \frac{5}{2} : \frac{5}{4} a^{5} - \frac{19}{8} a^{4} - \frac{29}{8} a^{3} + \frac{61}{8} a^{2} + \frac{7}{4} a - \frac{19}{8} : 1\right)$
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: \( 1620.3151885621828829270559259160438049 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 1.32957 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((a^5-4a^3+3a-2)\) \(79\) \(2\) \(I_{2}\) Non-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
\(2\) 2Cs
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 79.3-d consists of curves linked by isogenies of degrees dividing 12.

Base change

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

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