Properties

 Label 1.9.6t1.a.a Dimension $1$ Group $C_6$ Conductor $9$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$9$$$$\medspace = 3^{2}$$ Artin number field: Galois closure of $$\Q(\zeta_{9})$$ Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{9}(5,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

Defining polynomial

 $f(x)$ $=$ $x^{6} - x^{3} + 1$.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{2} + 16 x + 3$

Roots:
 $r_{ 1 }$ $=$ $11 a + 5 + \left(8 a + 14\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + \left(8 a + 7\right)\cdot 17^{3} + \left(a + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 a + 3 + \left(6 a + 5\right)\cdot 17 + \left(16 a + 6\right)\cdot 17^{2} + \left(13 a + 5\right)\cdot 17^{3} + \left(3 a + 3\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $10 a + 5 + \left(14 a + 12\right)\cdot 17 + \left(2 a + 4\right)\cdot 17^{2} + \left(5 a + 6\right)\cdot 17^{3} + \left(2 a + 12\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 4 }$ $=$ $7 a + 15 + \left(2 a + 16\right)\cdot 17 + \left(14 a + 9\right)\cdot 17^{2} + \left(11 a + 8\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 5 }$ $=$ $13 a + 7 + \left(10 a + 7\right)\cdot 17 + 16\cdot 17^{2} + \left(3 a + 2\right)\cdot 17^{3} + \left(13 a + 10\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 6 }$ $=$ $6 a + 16 + \left(8 a + 11\right)\cdot 17 + 3 a\cdot 17^{2} + \left(8 a + 3\right)\cdot 17^{3} + \left(15 a + 4\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2,3,6,5,4)$ $(1,6)(2,5)(3,4)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,6)(2,5)(3,4)$ $-1$ $1$ $3$ $(1,3,5)(2,6,4)$ $\zeta_{3}$ $1$ $3$ $(1,5,3)(2,4,6)$ $-\zeta_{3} - 1$ $1$ $6$ $(1,2,3,6,5,4)$ $\zeta_{3} + 1$ $1$ $6$ $(1,4,5,6,3,2)$ $-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.