Properties

Label 6.6.371293.1-79.3-d2
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

\({y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){x}{y}+\left(a^{5}-5a^{3}+6a+1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+5a^{2}-3a-4\right){x}^{2}+\left(-50a^{5}-22a^{4}+304a^{3}+108a^{2}-451a-112\right){x}-360a^{5}+56a^{4}+1956a^{3}+50a^{2}-2509a-581\)
sage: E = EllipticCurve([K([2,-3,-4,1,1,0]),K([-4,-3,5,4,-1,-1]),K([1,6,0,-5,0,1]),K([-112,-451,108,304,-22,-50]),K([-581,-2509,50,1956,56,-360])])
 
gp: E = ellinit([Polrev([2,-3,-4,1,1,0]),Polrev([-4,-3,5,4,-1,-1]),Polrev([1,6,0,-5,0,1]),Polrev([-112,-451,108,304,-22,-50]),Polrev([-581,-2509,50,1956,56,-360])], K);
 
magma: E := EllipticCurve([K![2,-3,-4,1,1,0],K![-4,-3,5,4,-1,-1],K![1,6,0,-5,0,1],K![-112,-451,108,304,-22,-50],K![-581,-2509,50,1956,56,-360]]);
 

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: \((-2581a^5-641a^4+12076a^3+431a^2-16295a+1497)\) = \((a^5-4a^3+3a-2)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 59091511031674153381441 \) = \(79^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1806310987189829549208270954886}{59091511031674153381441} a^{5} + \frac{1867280061295172198918453052615}{59091511031674153381441} a^{4} + \frac{4523353594532011391296947710931}{59091511031674153381441} a^{3} - \frac{1449083696217322076004722167442}{59091511031674153381441} a^{2} - \frac{2048522112959738678989772027522}{59091511031674153381441} a - \frac{353715935476625976028359928413}{59091511031674153381441} \)
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: $\left(\frac{11}{2} a^{5} - \frac{5}{4} a^{4} - \frac{113}{4} a^{3} - \frac{1}{4} a^{2} + \frac{65}{2} a + \frac{27}{4} : \frac{39}{8} a^{5} - \frac{185}{8} a^{3} - \frac{7}{2} a^{2} + 23 a + \frac{13}{2} : 1\right)$
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: \( 25.317424821284107545735248842438184452 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.32957 \)
Analytic order of Ш: \( 64 \) (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\((j)_{-}\)
\((a^5-4a^3+3a-2)\) \(79\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
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.