Properties

Base field \(\Q(\zeta_{9})^+\)
Label 3.3.81.1-17.3-a6
Conductor \((17,a^{2} - 2 a - 3)\)
Conductor norm \( 17 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
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

\( y^2 + \left(a^{2} + a - 2\right) x y + \left(a^{2} + a - 1\right) y = x^{3} + \left(a^{2} + a - 2\right) x^{2} + \left(-35 a^{2} + 102 a - 63\right) x - 639 a^{2} + 789 a + 777 \)
magma: E := ChangeRing(EllipticCurve([a^2 + a - 2, a^2 + a - 2, a^2 + a - 1, -35*a^2 + 102*a - 63, -639*a^2 + 789*a + 777]),K);
 
sage: E = EllipticCurve(K, [a^2 + a - 2, a^2 + a - 2, a^2 + a - 1, -35*a^2 + 102*a - 63, -639*a^2 + 789*a + 777])
 
gp (2.8): E = ellinit([a^2 + a - 2, a^2 + a - 2, a^2 + a - 1, -35*a^2 + 102*a - 63, -639*a^2 + 789*a + 777],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((17,a^{2} - 2 a - 3)\) = \( \left(a^{2} - 2 a - 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 17 \) = \( 17 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((17,a + 14,a^{2} + 8)\) = \( \left(a^{2} - 2 a - 3\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 17 \) = \( 17 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{5036213937953050641}{17} a^{2} - \frac{7715927403056521557}{17} a - \frac{3287155192616633274}{17} \)
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: \(\Z/2\Z\)
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: $\left(-5 a^{2} + \frac{17}{4} a + \frac{17}{2} : -\frac{35}{8} a^{2} + \frac{25}{8} a + \frac{75}{8} : 1\right)$
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\((j)_{-}\)
\( \left(a^{2} - 2 a - 3\right) \) \(17\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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.1.2

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 17.3-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.