Properties

Base field 6.6.434581.1
Label 6.6.434581.1-49.1-b1
Conductor \((7,a^{5} - 4 a^{4} + 11 a^{2} - 3 a - 4)\)
Conductor norm \( 49 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field 6.6.434581.1

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 5*x^3 + 4*x^2 - 2*x - 1)
 
gp: K = nfinit(a^6 - 2*a^5 - 4*a^4 + 5*a^3 + 4*a^2 - 2*a - 1);
 

Weierstrass equation

\( y^2 + \left(2 a^{5} - 4 a^{4} - 6 a^{3} + 7 a^{2} + 2 a - 1\right) x y + \left(a^{5} - a^{4} - 5 a^{3} + 4 a + 2\right) y = x^{3} + \left(2 a^{5} - 5 a^{4} - 5 a^{3} + 11 a^{2} + a - 3\right) x^{2} + \left(-a^{5} + a^{4} + 7 a^{3} - 3 a^{2} - 11 a + 1\right) x + 2 a^{5} - 5 a^{4} - 5 a^{3} + 12 a^{2} - 4 \)
magma: E := ChangeRing(EllipticCurve([2*a^5 - 4*a^4 - 6*a^3 + 7*a^2 + 2*a - 1, 2*a^5 - 5*a^4 - 5*a^3 + 11*a^2 + a - 3, a^5 - a^4 - 5*a^3 + 4*a + 2, -a^5 + a^4 + 7*a^3 - 3*a^2 - 11*a + 1, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 4]),K);
 
sage: E = EllipticCurve(K, [2*a^5 - 4*a^4 - 6*a^3 + 7*a^2 + 2*a - 1, 2*a^5 - 5*a^4 - 5*a^3 + 11*a^2 + a - 3, a^5 - a^4 - 5*a^3 + 4*a + 2, -a^5 + a^4 + 7*a^3 - 3*a^2 - 11*a + 1, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 4])
 
gp: E = ellinit([2*a^5 - 4*a^4 - 6*a^3 + 7*a^2 + 2*a - 1, 2*a^5 - 5*a^4 - 5*a^3 + 11*a^2 + a - 3, a^5 - a^4 - 5*a^3 + 4*a + 2, -a^5 + a^4 + 7*a^3 - 3*a^2 - 11*a + 1, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 4],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((7,a^{5} - 4 a^{4} + 11 a^{2} - 3 a - 4)\) = \( \left(-a^{5} + 4 a^{4} - 11 a^{2} + 3 a + 4\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 49 \) = \( 49 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((7,7 a^{4} - 14 a^{3} - 21 a^{2} + 21 a + 7,7 a^{5} - 21 a^{4} - 14 a^{3} + 56 a^{2} - 21,7 a,7 a^{5} - 14 a^{4} - 21 a^{3} + 21 a^{2} + 7 a,7 a^{5} - 14 a^{4} - 21 a^{3} + 28 a^{2} - 7)\) = \( \left(-a^{5} + 4 a^{4} - 11 a^{2} + 3 a + 4\right)^{3} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 117649 \) = \( 49^{3} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{2333735}{7} a^{5} - \frac{1536755}{7} a^{4} - 1654035 a^{3} - \frac{3465769}{7} a^{2} + \frac{4986743}{7} a + \frac{1824569}{7} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: Trivial
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a^{5} + 4 a^{4} - 11 a^{2} + 3 a + 4\right) \) \(49\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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 49.1-b 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.