↑

Properties

Label 6.6.703493.1-64.1-a1
Base field 6.6.703493.1
Conductor norm \( 64 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field 6.6.703493.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
 
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-6a^{3}+8a+1\right){x}{y}+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+7a-1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-5a^{2}-7a+2\right){x}^{2}+\left(-4a^{5}+2a^{4}+23a^{3}-12a^{2}-22a+3\right){x}-7a^{5}+4a^{4}+39a^{3}-20a^{2}-41a+5\)
sage: E = EllipticCurve([K([1,8,0,-6,0,1]),K([2,-7,-5,6,1,-1]),K([-1,7,5,-6,-1,1]),K([3,-22,-12,23,2,-4]),K([5,-41,-20,39,4,-7])])
 
gp: E = ellinit([Polrev([1,8,0,-6,0,1]),Polrev([2,-7,-5,6,1,-1]),Polrev([-1,7,5,-6,-1,1]),Polrev([3,-22,-12,23,2,-4]),Polrev([5,-41,-20,39,4,-7])], K);
 
magma: E := EllipticCurve([K![1,8,0,-6,0,1],K![2,-7,-5,6,1,-1],K![-1,7,5,-6,-1,1],K![3,-22,-12,23,2,-4],K![5,-41,-20,39,4,-7]]);
 

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: \((-2)\) = \((2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -64 \) = \(-64\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 146989275419572 a^{5} - \frac{143700390471519}{2} a^{4} - 843525494076395 a^{3} + \frac{684313694606955}{2} a^{2} + 811041701707335 a - \frac{194534795299797}{2} \)
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: \( 1363.2075330443049739109053164254180395 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.62530 \)
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\((j)_{-}\)
\((2)\) \(64\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 64.1-a consists of this curve only.

Base change

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

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