Properties

Label 6.6.453789.1-1.1-a3
Base field \(\Q(\zeta_{21})^+\)
Conductor norm \( 1 \)
CM yes (\(-147\))
Base change yes
Q-curve yes
Torsion order \( 7 \)
Rank \( 0 \)

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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

\({y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(131a^{5}-79a^{4}-814a^{3}+488a^{2}+1133a-800\right){x}-1218a^{5}+802a^{4}+7465a^{3}-4959a^{2}-10420a+7417\)
sage: E = EllipticCurve([K([0,0,0,0,0,0]),K([1,1,-3,0,1,0]),K([-2,2,1,-4,0,1]),K([-800,1133,488,-814,-79,131]),K([7417,-10420,-4959,7465,802,-1218])])
 
gp: E = ellinit([Polrev([0,0,0,0,0,0]),Polrev([1,1,-3,0,1,0]),Polrev([-2,2,1,-4,0,1]),Polrev([-800,1133,488,-814,-79,131]),Polrev([7417,-10420,-4959,7465,802,-1218])], K);
 
magma: E := EllipticCurve([K![0,0,0,0,0,0],K![1,1,-3,0,1,0],K![-2,2,1,-4,0,1],K![-800,1133,488,-814,-79,131],K![7417,-10420,-4959,7465,802,-1218]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1 \) = 1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 7604567359488000 a^{5} + 7604567359488000 a^{4} - 38022836797440000 a^{3} - 30418269437952000 a^{2} + 45627404156928000 a - 6017401737216000 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-147})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/7\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{12}{7} a^{5} - 2 a^{4} - \frac{64}{7} a^{3} + \frac{82}{7} a^{2} + \frac{86}{7} a - \frac{62}{7} : \frac{143}{7} a^{5} - \frac{94}{7} a^{4} - \frac{853}{7} a^{3} + \frac{618}{7} a^{2} + \frac{1199}{7} a - 125 : 1\right)$
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: \( 22162.259801148018040327406817706972594 \)
Tamagawa product: \( 1 \)
Torsion order: \(7\)
Leading coefficient: \( 0.671415 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.1[3]

For all other primes \(p\), the image is a Borel subgroup if \(p=3\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 7, 21, 49 and 147.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 147.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q(\sqrt{21}) \) a curve with conductor norm 49 (not in the database)