Properties

Label 4.4.4913.1-17.1-a8
Base field 4.4.4913.1
Conductor \((1/2a^3-2a-3/2)\)
Conductor norm \( 17 \)
CM no
Base change yes: 17.a4,289.a4
Q-curve yes
Torsion order \( 16 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Base field 4.4.4913.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -6, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, 1, -6, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/2a^3-2a-3/2)\) = \((1/2a^3-2a-3/2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 17 \) = \(17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((17)\) = \((1/2a^3-2a-3/2)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 83521 \) = \(17^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{35937}{17} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(a^{3} - a^{2} - 5 a + 2 : -2 a^{3} + 2 a^{2} + 10 a - 5 : 1\right)$
Height \(0.758814044432583\)
Torsion structure: \(\Z/2\Z\times\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{5}{4} a : -\frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{5}{8} a - \frac{1}{2} : 1\right)$ $\left(\frac{3}{2} a^{3} - 2 a^{2} - 8 a + \frac{9}{2} : 3 a^{3} - 4 a^{2} - 17 a + 8 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.758814044432583 \)
Period: \( 1466.52940633504 \)
Tamagawa product: \( 4 \)
Torsion order: \(16\)
Leading coefficient: \( 0.992276649128961 \)
Analytic order of Ш: \( 1 \) (rounded)

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\((j)_{-}\)
\((1/2a^3-2a-3/2)\) \(17\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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 17.a4, 289.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.