Properties

Label 6.6.1241125.1-29.1-a1
Base field 6.6.1241125.1
Conductor \((a^4-a^3-5a^2+3a+4)\)
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 7 \)
Rank \( 1 \)

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Base field 6.6.1241125.1

Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, 7, 11, -2, -7, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-3\right){x}{y}+\left(-a^{5}+a^{4}+6a^{3}-3a^{2}-7a-2\right){y}={x}^{3}+\left(-2a^{5}+a^{4}+13a^{3}-2a^{2}-19a-5\right){x}^{2}+\left(-4a^{5}+2a^{4}+26a^{3}-5a^{2}-38a-9\right){x}-23a^{5}+11a^{4}+155a^{3}-28a^{2}-237a-49\)
sage: E = EllipticCurve([K([-3,0,1,0,0,0]),K([-5,-19,-2,13,1,-2]),K([-2,-7,-3,6,1,-1]),K([-9,-38,-5,26,2,-4]),K([-49,-237,-28,155,11,-23])])
 
gp: E = ellinit([Pol(Vecrev([-3,0,1,0,0,0])),Pol(Vecrev([-5,-19,-2,13,1,-2])),Pol(Vecrev([-2,-7,-3,6,1,-1])),Pol(Vecrev([-9,-38,-5,26,2,-4])),Pol(Vecrev([-49,-237,-28,155,11,-23]))], K);
 
magma: E := EllipticCurve([K![-3,0,1,0,0,0],K![-5,-19,-2,13,1,-2],K![-2,-7,-3,6,1,-1],K![-9,-38,-5,26,2,-4],K![-49,-237,-28,155,11,-23]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-a^3-5a^2+3a+4)\) = \((a^4-a^3-5a^2+3a+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29 \) = \(29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^5+a^4+7a^3-4a^2-11a)\) = \((a^4-a^3-5a^2+3a+4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 29 \) = \(29\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{808367}{29} a^{5} - \frac{232409}{29} a^{4} + \frac{3911969}{29} a^{3} + \frac{1920337}{29} a^{2} - \frac{569967}{29} a - \frac{293942}{29} \)
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 $\left(a^{5} - 7 a^{3} - 2 a^{2} + 11 a + 7 : 12 a^{5} - 4 a^{4} - 82 a^{3} + 2 a^{2} + 128 a + 45 : 1\right)$
Height \(0.220715821358718\)
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: $\left(0 : 4 a^{5} - 2 a^{4} - 27 a^{3} + 5 a^{2} + 41 a + 9 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: not available
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.220715821358718 \)
Period: not available
Tamagawa product: \( 1 \)
Torsion order: \(7\)
Leading coefficient: not available
Analytic order of Ш: not available

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\((j)_{-}\)
\((a^4-a^3-5a^2+3a+4)\) \(29\) \(1\) \(I_{1}\) 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
\(7\) 7B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 29.1-a consists of curves linked by isogenies of degree 7.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.