Properties

Label 6.6.453789.1-41.6-c1
Base field \(\Q(\zeta_{21})^+\)
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\zeta_{21})^+\)

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -8, 8, 6, -6, -1, 1]))
 
gp: K = nfinit(Polrev([1, -8, 8, 6, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 8, 6, -6, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-4a^{3}+2a^{2}+3a-4\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+6a^{2}+7a-7\right){x}^{2}+\left(-a^{5}+3a^{4}+11a^{3}-9a^{2}-22a+5\right){x}+6a^{5}+2a^{4}-31a^{3}-2a^{2}+37a-6\)
sage: E = EllipticCurve([K([-4,3,2,-4,0,1]),K([-7,7,6,-6,-1,1]),K([0,-3,0,1,0,0]),K([5,-22,-9,11,3,-1]),K([-6,37,-2,-31,2,6])])
 
gp: E = ellinit([Polrev([-4,3,2,-4,0,1]),Polrev([-7,7,6,-6,-1,1]),Polrev([0,-3,0,1,0,0]),Polrev([5,-22,-9,11,3,-1]),Polrev([-6,37,-2,-31,2,6])], K);
 
magma: E := EllipticCurve([K![-4,3,2,-4,0,1],K![-7,7,6,-6,-1,1],K![0,-3,0,1,0,0],K![5,-22,-9,11,3,-1],K![-6,37,-2,-31,2,6]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^5-a^4+5a^3+3a^2-5a-1)\) = \((-a^5-a^4+5a^3+3a^2-5a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \(41\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^5+6a^3-9a+2)\) = \((-a^5-a^4+5a^3+3a^2-5a-1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -41 \) = \(-41\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{21474379287}{41} a^{5} - \frac{6745914900}{41} a^{4} - \frac{134652336687}{41} a^{3} + \frac{34557144345}{41} a^{2} + \frac{199103938866}{41} a - \frac{30078826092}{41} \)
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 $\left(a + 1 : a + 2 : 1\right)$
Height \(0.0041762855719201394771630882797172046147\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.0041762855719201394771630882797172046147 \)
Period: \( 59892.466146077458345329072565085321460 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.22786 \)
Analytic order of Ш: \( 1 \) (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\((j)_{-}\)
\((-a^5-a^4+5a^3+3a^2-5a-1)\) \(41\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 41.6-c consists of curves linked by isogenies of degree 3.

Base change

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

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