Properties

Base field 4.4.4913.1
Label 4.4.4913.1-17.1-a8
Conductor \((17,\frac{1}{2} a^{3} - a^{2} - a + \frac{3}{2})\)
Conductor norm \( 17 \)
CM no
base-change yes: 17.a4,289.a4
Q-curve yes
Torsion order \( 16 \)
Rank not available

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Base field 4.4.4913.1

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

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

Weierstrass equation

\( y^2 + x y + y = x^{3} - x^{2} - x \)
magma: E := ChangeRing(EllipticCurve([1, -1, 1, -1, 0]),K);
sage: E = EllipticCurve(K, [1, -1, 1, -1, 0])
gp (2.8): E = ellinit([1, -1, 1, -1, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((17,\frac{1}{2} a^{3} - a^{2} - a + \frac{3}{2})\) = \( \left(\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 17 \) = \( 17 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((17,17 a,\frac{17}{2} a^{3} - 51 a - \frac{51}{2},-\frac{17}{2} a^{3} + 17 a^{2} + 34 a - \frac{51}{2})\) = \( \left(\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 83521 \) = \( 17^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{35937}{17} \)
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: \(\Z/2\Z\times\Z/8\Z\)
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: $\left(\frac{3}{2} a^{3} - 2 a^{2} - 8 a + \frac{9}{2} : 3 a^{3} - 4 a^{2} - 17 a + 8 : 1\right)$,$\left(\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{5}{4} a : -\frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{5}{8} a - \frac{1}{2} : 1\right)$
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\((j)_{-}\)
\( \left(\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\right) \) \(17\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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, 4 and 8.
Its isogeny class 17.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 17.a4, 289.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.