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Properties

Base field 6.6.485125.1
Label 6.6.485125.1-31.1-a3
Conductor \((31,-a^{4} + 4 a^{2} + a - 3)\)
Conductor norm \( 31 \)
CM no
base-change no
Q-curve no
Torsion order \( 8 \)
Rank not available

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 8*x^3 + 2*x^2 - 5*x + 1)
 
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 8*a^3 + 2*a^2 - 5*a + 1);
 

Weierstrass equation

\( y^2 + \left(2 a^{5} - 2 a^{4} - 9 a^{3} + 7 a^{2} + 7 a - 2\right) x y + \left(-a^{5} + 2 a^{4} + 4 a^{3} - 6 a^{2} - 2 a + 1\right) y = x^{3} + \left(-2 a^{5} + 3 a^{4} + 8 a^{3} - 9 a^{2} - 4 a + 3\right) x^{2} + \left(9 a^{5} - 40 a^{4} + 38 a^{3} + 32 a^{2} - 35 a + 2\right) x - 85 a^{5} + 329 a^{4} - 255 a^{3} - 243 a^{2} + 270 a - 47 \)
magma: E := ChangeRing(EllipticCurve([2*a^5 - 2*a^4 - 9*a^3 + 7*a^2 + 7*a - 2, -2*a^5 + 3*a^4 + 8*a^3 - 9*a^2 - 4*a + 3, -a^5 + 2*a^4 + 4*a^3 - 6*a^2 - 2*a + 1, 9*a^5 - 40*a^4 + 38*a^3 + 32*a^2 - 35*a + 2, -85*a^5 + 329*a^4 - 255*a^3 - 243*a^2 + 270*a - 47]),K);
 
sage: E = EllipticCurve(K, [2*a^5 - 2*a^4 - 9*a^3 + 7*a^2 + 7*a - 2, -2*a^5 + 3*a^4 + 8*a^3 - 9*a^2 - 4*a + 3, -a^5 + 2*a^4 + 4*a^3 - 6*a^2 - 2*a + 1, 9*a^5 - 40*a^4 + 38*a^3 + 32*a^2 - 35*a + 2, -85*a^5 + 329*a^4 - 255*a^3 - 243*a^2 + 270*a - 47])
 
gp (2.8): E = ellinit([2*a^5 - 2*a^4 - 9*a^3 + 7*a^2 + 7*a - 2, -2*a^5 + 3*a^4 + 8*a^3 - 9*a^2 - 4*a + 3, -a^5 + 2*a^4 + 4*a^3 - 6*a^2 - 2*a + 1, 9*a^5 - 40*a^4 + 38*a^3 + 32*a^2 - 35*a + 2, -85*a^5 + 329*a^4 - 255*a^3 - 243*a^2 + 270*a - 47],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((31,-a^{4} + 4 a^{2} + a - 3)\) = \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 31 \) = \( 31 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((887503681,a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a + 630753110,-a^{5} + 2 a^{4} + 4 a^{3} - 7 a^{2} - 3 a + 390826567,a^{5} - a^{4} - 4 a^{3} + 3 a^{2} + 2 a + 873930656,a + 81304435,-a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a + 523347266)\) = \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right)^{6} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 887503681 \) = \( 31^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{1460546550652965787030}{887503681} a^{5} + \frac{180239481432461299101}{887503681} a^{4} - \frac{5459549052609937959965}{887503681} a^{3} + \frac{91383042197658996772}{887503681} a^{2} + \frac{3115193703304519977774}{887503681} a - \frac{687821042793429817941}{887503681} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): 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()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: not available

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/4\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generators: $\left(2 a^{4} - 4 a^{3} - 3 a^{2} + 4 a + 1 : 3 a^{5} - 9 a^{4} + 4 a^{3} + 7 a^{2} - 7 a - 1 : 1\right)$,$\left(-a^{3} + 2 a^{2} - a - 3 : 4 a^{5} - 4 a^{4} - 17 a^{3} + 12 a^{2} + 14 a - 4 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[3]
 

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\((j)_{-}\)
\( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \) \(31\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 31.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

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