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Properties

Label 6.6.1279733.1-7.1-d5
Base field 6.6.1279733.1
Conductor norm \( 7 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field 6.6.1279733.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -11, 10, 10, -6, -2, 1]))
 
gp: K = nfinit(Polrev([-1, -11, 10, 10, -6, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 10, 10, -6, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-a^{4}-4a^{3}+3a^{2}+2a-2\right){x}{y}+\left(a^{4}-a^{3}-3a^{2}+2a+1\right){y}={x}^{3}+\left(-a^{5}+3a^{4}+3a^{3}-12a^{2}-a+6\right){x}^{2}+\left(-5a^{5}+61a^{3}-31a^{2}-139a+1\right){x}-44a^{5}-33a^{4}+383a^{3}+59a^{2}-696a-54\)
sage: E = EllipticCurve([K([-2,2,3,-4,-1,1]),K([6,-1,-12,3,3,-1]),K([1,2,-3,-1,1,0]),K([1,-139,-31,61,0,-5]),K([-54,-696,59,383,-33,-44])])
 
gp: E = ellinit([Polrev([-2,2,3,-4,-1,1]),Polrev([6,-1,-12,3,3,-1]),Polrev([1,2,-3,-1,1,0]),Polrev([1,-139,-31,61,0,-5]),Polrev([-54,-696,59,383,-33,-44])], K);
 
magma: E := EllipticCurve([K![-2,2,3,-4,-1,1],K![6,-1,-12,3,3,-1],K![1,2,-3,-1,1,0],K![1,-139,-31,61,0,-5],K![-54,-696,59,383,-33,-44]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-2a^3-3a^2+6a-1)\) = \((a^4-2a^3-3a^2+6a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 7 \) = \(7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^5+2a^4-11a^3-7a^2+22a+3)\) = \((a^4-2a^3-3a^2+6a-1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2401 \) = \(7^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{203442160742265836111311}{49} a^{5} + \frac{754939116457579934341120}{49} a^{4} - \frac{70918941964796608216091}{49} a^{3} - \frac{1913091408632068298504203}{49} a^{2} + \frac{1238551091877646917232536}{49} a + \frac{118914358788506027626050}{49} \)
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: \(1\)
Generator $\left(\frac{3}{13} a^{5} + \frac{56}{13} a^{4} - \frac{220}{13} a^{3} - \frac{96}{13} a^{2} + \frac{578}{13} a - \frac{12}{13} : \frac{3370}{169} a^{5} - \frac{6226}{169} a^{4} - \frac{14034}{169} a^{3} + \frac{17009}{169} a^{2} + \frac{17791}{169} a + \frac{798}{169} : 1\right)$
Height \(2.0221738809634743375006794661233233828\)
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: $\left(2 a^{5} - 7 a^{4} - 4 a^{3} + 23 a^{2} + a - 1 : -3 a^{5} + 3 a^{4} + 9 a^{3} - 6 a^{2} + 4 a - 1 : 1\right)$ $\left(-\frac{1}{4} a^{5} - a^{4} + \frac{11}{4} a^{3} + \frac{15}{4} a^{2} - 5 a - \frac{11}{2} : \frac{13}{4} a^{5} - \frac{11}{8} a^{4} - 15 a^{3} + \frac{11}{4} a^{2} + \frac{23}{2} a - \frac{39}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.0221738809634743375006794661233233828 \)
Period: \( 61.213009970393019701714240919659907123 \)
Tamagawa product: \( 4 \)
Torsion order: \(4\)
Leading coefficient: \( 2.62611 \)
Analytic order of Ш: \( 16 \) (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\((j)_{-}\)
\((a^4-2a^3-3a^2+6a-1)\) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 7.1-d consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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