Properties

Base field \(\Q(\zeta_{9})^+\)
Label 3.3.81.1-17.1-a8
Conductor \((17,-2 a^{2} + a + 3)\)
Conductor norm \( 17 \)
CM no
base-change no
Q-curve no
Torsion order \( 6 \)
Rank not available

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Base field \(\Q(\zeta_{9})^+\)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - 3*x - 1)
 
gp (2.8): K = nfinit(a^3 - 3*a - 1);
 

Weierstrass equation

\( y^2 + x y + \left(a^{2} + a - 2\right) y = x^{3} + \left(a^{2} - 2\right) x^{2} + \left(-311 a^{2} + 119 a + 874\right) x + 1414 a^{2} - 463 a - 4127 \)
magma: E := ChangeRing(EllipticCurve([1, a^2 - 2, a^2 + a - 2, -311*a^2 + 119*a + 874, 1414*a^2 - 463*a - 4127]),K);
 
sage: E = EllipticCurve(K, [1, a^2 - 2, a^2 + a - 2, -311*a^2 + 119*a + 874, 1414*a^2 - 463*a - 4127])
 
gp (2.8): E = ellinit([1, a^2 - 2, a^2 + a - 2, -311*a^2 + 119*a + 874, 1414*a^2 - 463*a - 4127],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((17,-2 a^{2} + a + 3)\) = \( \left(2 a^{2} - a - 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 17 \) = \( 17 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((582622237229761,a + 527282828196502,a^{2} + 404039907522171)\) = \( \left(2 a^{2} - a - 3\right)^{12} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 582622237229761 \) = \( 17^{12} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{4075073731548124101}{582622237229761} a^{2} - \frac{3433122709460467011}{582622237229761} a - \frac{810676269288190764}{582622237229761} \)
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: \(\Z/6\Z\)
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: $\left(-32 a^{2} + 12 a + 90 : -278 a^{2} + 95 a + 802 : 1\right)$
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\((j)_{-}\)
\( \left(2 a^{2} - a - 3\right) \) \(17\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.1

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 17.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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