Properties

Base field 6.6.434581.1
Label 6.6.434581.1-43.2-b1
Conductor \((43,a^{4} - a^{3} - 5 a^{2} + 4)\)
Conductor norm \( 43 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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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 \(1\).

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);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
 

Weierstrass equation

\( y^2 + \left(3 a^{5} - 6 a^{4} - 10 a^{3} + 12 a^{2} + 4 a - 3\right) x y + \left(a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} - 1\right) y = x^{3} + \left(a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + a - 3\right) x^{2} + \left(-2 a^{5} - 2 a^{4} + 13 a^{3} + 19 a^{2} - 4 a - 9\right) x - 13 a^{5} + 16 a^{4} + 56 a^{3} - 13 a^{2} - 25 a + 4 \)
sage: E = EllipticCurve(K, [3*a^5 - 6*a^4 - 10*a^3 + 12*a^2 + 4*a - 3, a^5 - 3*a^4 - 2*a^3 + 8*a^2 + a - 3, a^5 - 2*a^4 - 3*a^3 + 4*a^2 - 1, -2*a^5 - 2*a^4 + 13*a^3 + 19*a^2 - 4*a - 9, -13*a^5 + 16*a^4 + 56*a^3 - 13*a^2 - 25*a + 4])
 
gp: E = ellinit([3*a^5 - 6*a^4 - 10*a^3 + 12*a^2 + 4*a - 3, a^5 - 3*a^4 - 2*a^3 + 8*a^2 + a - 3, a^5 - 2*a^4 - 3*a^3 + 4*a^2 - 1, -2*a^5 - 2*a^4 + 13*a^3 + 19*a^2 - 4*a - 9, -13*a^5 + 16*a^4 + 56*a^3 - 13*a^2 - 25*a + 4],K)
 
magma: E := ChangeRing(EllipticCurve([3*a^5 - 6*a^4 - 10*a^3 + 12*a^2 + 4*a - 3, a^5 - 3*a^4 - 2*a^3 + 8*a^2 + a - 3, a^5 - 2*a^4 - 3*a^3 + 4*a^2 - 1, -2*a^5 - 2*a^4 + 13*a^3 + 19*a^2 - 4*a - 9, -13*a^5 + 16*a^4 + 56*a^3 - 13*a^2 - 25*a + 4]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((43,a^{4} - a^{3} - 5 a^{2} + 4)\) = \( \left(a^{5} - 3 a^{4} - a^{3} + 7 a^{2} - 3 a - 4\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 43 \) = \( 43 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((1849,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 717,a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 1365,a + 322,a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a + 573,a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} + 111)\) = \( \left(a^{5} - 3 a^{4} - a^{3} + 7 a^{2} - 3 a - 4\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 1849 \) = \( 43^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{35119198}{1849} a^{5} + \frac{13127784}{1849} a^{4} + \frac{79757853}{1849} a^{3} - \frac{41392973}{1849} a^{2} - \frac{11112889}{1849} a + \frac{3697481}{1849} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

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

Torsion subgroup

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

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\((j)_{-}\)
\( \left(a^{5} - 3 a^{4} - a^{3} + 7 a^{2} - 3 a - 4\right) \) \(43\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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 43.2-b consists of this curve only.

Base change

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