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Properties

Label 6.6.485125.1-1.1-a1
Base field 6.6.485125.1
Conductor norm \( 1 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.485125.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
 
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-a^{4}-4a^{3}+3a^{2}+3a-1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-7a-2\right){x}^{2}+\left(-126a^{5}-117a^{4}+741a^{3}+702a^{2}-1132a-1114\right){x}-3781a^{5}-2012a^{4}+17879a^{3}+9699a^{2}-19472a-13392\)
sage: E = EllipticCurve([K([-1,3,3,-4,-1,1]),K([-2,-7,0,6,0,-1]),K([-1,1,1,0,0,0]),K([-1114,-1132,702,741,-117,-126]),K([-13392,-19472,9699,17879,-2012,-3781])])
 
gp: E = ellinit([Polrev([-1,3,3,-4,-1,1]),Polrev([-2,-7,0,6,0,-1]),Polrev([-1,1,1,0,0,0]),Polrev([-1114,-1132,702,741,-117,-126]),Polrev([-13392,-19472,9699,17879,-2012,-3781])], K);
 
magma: E := EllipticCurve([K![-1,3,3,-4,-1,1],K![-2,-7,0,6,0,-1],K![-1,1,1,0,0,0],K![-1114,-1132,702,741,-117,-126],K![-13392,-19472,9699,17879,-2012,-3781]]);
 

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: \( 1606219307040269000685664528235176250888 a^{5} - 505201255314545228245706103526492515760 a^{4} - 7276379701134851706189760667243713057765 a^{3} + 585621608998000505707418880591397665054 a^{2} + 4199487325941333963885321109586284787419 a - 952979041145067887798527201046030804711 \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.030826936445874590795690667924708465996 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.647999 \)
Analytic order of Ш: \( 14641 \) (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
\(11\) 11B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 11 and 33.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 33.

Base change

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

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