Properties

Label 6.6.1312625.1-4.1-a1
Base field 6.6.1312625.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 6.6.1312625.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 7 x^{3} + 12 x^{2} - 12 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -12, 12, 7, -7, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -12, 12, 7, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -12, 12, 7, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-6a^{3}+9a\right){x}{y}+\left(a^{5}-6a^{3}+a^{2}+9a-3\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-8a+2\right){x}^{2}+\left(366a^{5}+192a^{4}-2302a^{3}-911a^{2}+3138a+245\right){x}+4708a^{5}+2285a^{4}-29436a^{3}-10909a^{2}+39749a+3127\)
sage: E = EllipticCurve([K([0,9,0,-6,0,1]),K([2,-8,-4,6,1,-1]),K([-3,9,1,-6,0,1]),K([245,3138,-911,-2302,192,366]),K([3127,39749,-10909,-29436,2285,4708])])
 
gp: E = ellinit([Polrev([0,9,0,-6,0,1]),Polrev([2,-8,-4,6,1,-1]),Polrev([-3,9,1,-6,0,1]),Polrev([245,3138,-911,-2302,192,366]),Polrev([3127,39749,-10909,-29436,2285,4708])], K);
 
magma: E := EllipticCurve([K![0,9,0,-6,0,1],K![2,-8,-4,6,1,-1],K![-3,9,1,-6,0,1],K![245,3138,-911,-2302,192,366],K![3127,39749,-10909,-29436,2285,4708]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^5+6a^3+a^2-8a-2)\) = \((-a^5+6a^3+a^2-8a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^5+a^4-9a^3-6a^2+16a+4)\) = \((-a^5+6a^3+a^2-8a-2)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4096 \) = \(4^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{10576507188547681123}{64} a^{5} - \frac{2616146338735194005}{32} a^{4} + \frac{16553698917074597961}{16} a^{3} + \frac{24936296834825976631}{64} a^{2} - \frac{89645580404194791495}{64} a - \frac{3537982172908461591}{32} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{33}{4} a^{5} + \frac{5}{2} a^{4} - 51 a^{3} - \frac{49}{4} a^{2} + 68 a + \frac{13}{4} : -\frac{41}{8} a^{5} - \frac{17}{8} a^{4} + \frac{263}{8} a^{3} + \frac{19}{2} a^{2} - \frac{191}{4} a - \frac{21}{8} : 1\right)$
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: \( 2799.3372214788879792328427346169253742 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.22167 \)
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\((j)_{-}\)
\((-a^5+6a^3+a^2-8a-2)\) \(4\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 4.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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