Properties

Base field 3.3.316.1
Label 3.3.316.1-32.1-a5
Conductor \((4,2 a^{2} - 8)\)
Conductor norm \( 32 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
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

\( y^2 + a x y + a y = x^{3} + \left(-a^{2} + a + 4\right) x^{2} + \left(26100179 a^{2} - 13815259 a - 110903333\right) x + 106175166175 a^{2} - 56200275854 a - 451153202737 \)
magma: E := ChangeRing(EllipticCurve([a, -a^2 + a + 4, a, 26100179*a^2 - 13815259*a - 110903333, 106175166175*a^2 - 56200275854*a - 451153202737]),K);
sage: E = EllipticCurve(K, [a, -a^2 + a + 4, a, 26100179*a^2 - 13815259*a - 110903333, 106175166175*a^2 - 56200275854*a - 451153202737])
gp (2.8): E = ellinit([a, -a^2 + a + 4, a, 26100179*a^2 - 13815259*a - 110903333, 106175166175*a^2 - 56200275854*a - 451153202737],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()
\(\mathfrak{D}\) = \((32,32 a,4 a^{2} + 8 a + 8)\) = \( \left(-a\right)^{10} \cdot \left(-a + 1\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 4096 \) = \( 2^{12} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{58508563841049}{4} a^{2} + \frac{15484776480827}{2} a + \frac{124305555260903}{2} \)
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: \(\Z/2\Z\)
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]
Generator: $\left(\frac{5707}{4} a^{2} - 755 a - 6062 : -\frac{2687}{8} a^{2} + 177 a + \frac{5707}{4} : 1\right)$
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\((j)_{-}\)
\( \left(-a\right) \) \(2\) \(2\) \(I_{2}^*\) Additive \(-1\) \(4\) \(10\) \(0\)
\( \left(-a + 1\right) \) \(2\) \(2\) \(I_{2}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
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.