Properties

Base field 3.3.316.1
Label 3.3.316.1-32.1-a3
Conductor \((4,2 a^{2} - 8)\)
Conductor norm \( 32 \)
CM no
base-change no
Q-curve no
Torsion order \( 4 \)
Rank not available

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Base field 3.3.316.1

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

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

Weierstrass equation

\( y^2 + a x y + a y = x^{3} + \left(a^{2} + a - 2\right) x^{2} + \left(13 a^{2} - 3 a - 50\right) x + 19 a^{2} - 7 a - 76 \)
magma: E := ChangeRing(EllipticCurve([a, a^2 + a - 2, a, 13*a^2 - 3*a - 50, 19*a^2 - 7*a - 76]),K);
 
sage: E = EllipticCurve(K, [a, a^2 + a - 2, a, 13*a^2 - 3*a - 50, 19*a^2 - 7*a - 76])
 
gp (2.8): E = ellinit([a, a^2 + a - 2, a, 13*a^2 - 3*a - 50, 19*a^2 - 7*a - 76],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((4,2 a^{2} - 8)\) = \( \left(-a\right)^{4} \cdot \left(-a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 32 \) = \( 2^{5} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((4,4 a,4 a^{2} - 12)\) = \( \left(-a\right)^{4} \cdot \left(-a + 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 64 \) = \( 2^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{1401505}{4} a^{2} + \frac{928349}{2} a - \frac{584947}{2} \)
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: \(\Z/2\Z\times\Z/2\Z\)
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: $\left(-\frac{1}{2} a - \frac{3}{2} : \frac{1}{4} a^{2} + \frac{1}{4} a : 1\right)$,$\left(a^{2} - a - 5 : 1 : 1\right)$
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\((j)_{-}\)
\( \left(-a\right) \) \(2\) \(1\) \(II\) Additive \(-1\) \(4\) \(4\) \(0\)
\( \left(-a + 1\right) \) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 32.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

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