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Properties

Label 6.6.434581.1-1.1-a4
Base field 6.6.434581.1
Conductor norm \( 1 \)
CM no
Base change no
Q-curve no
Torsion order \( 13 \)
Rank \( 0 \)

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Base field 6.6.434581.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-4a^{4}-7a^{3}+8a^{2}+4a-1\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+3a+1\right){y}={x}^{3}+\left(a^{5}-3a^{4}-2a^{3}+8a^{2}-4\right){x}^{2}+\left(-3a^{5}+4a^{4}+15a^{3}-6a^{2}-17a-3\right){x}+3a^{5}-4a^{4}-15a^{3}+5a^{2}+16a+4\)
sage: E = EllipticCurve([K([-1,4,8,-7,-4,2]),K([-4,0,8,-2,-3,1]),K([1,3,-3,-2,1,0]),K([-3,-17,-6,15,4,-3]),K([4,16,5,-15,-4,3])])
 
gp: E = ellinit([Polrev([-1,4,8,-7,-4,2]),Polrev([-4,0,8,-2,-3,1]),Polrev([1,3,-3,-2,1,0]),Polrev([-3,-17,-6,15,4,-3]),Polrev([4,16,5,-15,-4,3])], K);
 
magma: E := EllipticCurve([K![-1,4,8,-7,-4,2],K![-4,0,8,-2,-3,1],K![1,3,-3,-2,1,0],K![-3,-17,-6,15,4,-3],K![4,16,5,-15,-4,3]]);
 

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: \( -23419364 a^{5} + 64630767 a^{4} + 44556569 a^{3} - 150905488 a^{2} + 21033374 a + 30775655 \)
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: \(\Z/13\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 a^{4} - 5 a^{2} + 2 : 6 a^{5} + 2 a^{4} - 15 a^{3} - 4 a^{2} + 7 a + 3 : 1\right)$
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: \( 71447.187681101328541463561259989660288 \)
Tamagawa product: \( 1 \)
Torsion order: \(13\)
Leading coefficient: \( 0.641303 \)
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
\(3\) 3B
\(13\) 13B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 13 and 39.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 39.

Base change

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

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