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Properties

Label 6.6.300125.1-64.1-a1
Base field 6.6.300125.1
Conductor norm \( 64 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.300125.1

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

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

Weierstrass equation

\({y}^2+\left(-2a^{5}+a^{4}+14a^{3}+4a^{2}-9a-2\right){x}{y}+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-6\right){y}={x}^{3}+\left(-3a^{5}+a^{4}+21a^{3}+8a^{2}-10a-1\right){x}^{2}+\left(10524a^{5}-2534a^{4}-75422a^{3}-36441a^{2}+45216a+13058\right){x}+342107a^{5}-79925a^{4}-2453847a^{3}-1199258a^{2}+1465409a+437699\)
sage: E = EllipticCurve([K([-2,-9,4,14,1,-2]),K([-1,-10,8,21,1,-3]),K([-6,-29,17,43,2,-6]),K([13058,45216,-36441,-75422,-2534,10524]),K([437699,1465409,-1199258,-2453847,-79925,342107])])
 
gp: E = ellinit([Polrev([-2,-9,4,14,1,-2]),Polrev([-1,-10,8,21,1,-3]),Polrev([-6,-29,17,43,2,-6]),Polrev([13058,45216,-36441,-75422,-2534,10524]),Polrev([437699,1465409,-1199258,-2453847,-79925,342107])], K);
 
magma: E := EllipticCurve([K![-2,-9,4,14,1,-2],K![-1,-10,8,21,1,-3],K![-6,-29,17,43,2,-6],K![13058,45216,-36441,-75422,-2534,10524],K![437699,1465409,-1199258,-2453847,-79925,342107]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 64 \) = \(64\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8192)\) = \((2)^{13}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 302231454903657293676544 \) = \(64^{13}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{5745702166029}{8192} \)
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: \( 0.025668910289160513700041352335409205574 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.33823 \)
Analytic order of Ш: \( 28561 \) (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\((j)_{-}\)
\((2)\) \(64\) \(1\) \(I_{13}\) Non-split multiplicative \(1\) \(1\) \(13\) \(13\)

Galois Representations

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

prime Image of Galois Representation
\(13\) 13B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 13.
Its isogeny class 64.1-a consists of curves linked by isogenies of degree 13.

Base change

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

Base field Curve
\(\Q(\sqrt{5}) \) a curve with conductor norm 9604 (not in the database)