Properties

Label 6.6.1387029.1-27.1-p2
Base field 6.6.1387029.1
Conductor \((a^5-3a^4-3a^3+8a^2+2a-2)\)
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Base field 6.6.1387029.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, -4, -1, 9, -2, -3, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(-a^{5}+3a^{4}+2a^{3}-7a^{2}+2\right){y}={x}^{3}+\left(3a^{5}-7a^{4}-9a^{3}+18a^{2}+4a-5\right){x}^{2}+\left(7a^{5}-34a^{4}+40a^{3}+16a^{2}-31a+10\right){x}+53a^{5}-232a^{4}+239a^{3}+81a^{2}-149a+29\)
sage: E = EllipticCurve([K([1,-4,0,1,0,0]),K([-5,4,18,-9,-7,3]),K([2,0,-7,2,3,-1]),K([10,-31,16,40,-34,7]),K([29,-149,81,239,-232,53])])
 
gp: E = ellinit([Pol(Vecrev([1,-4,0,1,0,0])),Pol(Vecrev([-5,4,18,-9,-7,3])),Pol(Vecrev([2,0,-7,2,3,-1])),Pol(Vecrev([10,-31,16,40,-34,7])),Pol(Vecrev([29,-149,81,239,-232,53]))], K);
 
magma: E := EllipticCurve([K![1,-4,0,1,0,0],K![-5,4,18,-9,-7,3],K![2,0,-7,2,3,-1],K![10,-31,16,40,-34,7],K![29,-149,81,239,-232,53]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-3a^4-3a^3+8a^2+2a-2)\) = \((a^5-3a^4-2a^3+8a^2+a-2)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(3^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-5a^5+17a^4+9a^3-54a^2+3a+23)\) = \((a^5-3a^4-2a^3+8a^2+a-2)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -177147 \) = \(-3^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 3426222102 a^{5} - 12812570435 a^{4} + 2623723832 a^{3} + 28894116691 a^{2} - 24794656162 a + 4632631517 \)
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: not available
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: not available
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: not available
Analytic order of Ш: not available

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\((j)_{-}\)
\((a^5-3a^4-2a^3+8a^2+a-2)\) \(3\) \(1\) \(II^{*}\) Additive \(1\) \(3\) \(11\) \(0\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 27.1-p consists of curves linked by isogenies of degree 3.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.