# Properties

 Label 1.117.6t1.a.a Dimension 1 Group $C_6$ Conductor $3^{2} \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $117= 3^{2} \cdot 13$ Artin number field: Splitting field of 6.0.562166163.1 defined by $f= x^{6} - 91 x^{3} + 2197$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{117}(74,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$
Roots:
 $r_{ 1 }$ $=$ $12 a + 2 + \left(10 a + 17\right)\cdot 23 + \left(15 a + 7\right)\cdot 23^{2} + \left(10 a + 4\right)\cdot 23^{3} + \left(16 a + 1\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 a + 12 + \left(20 a + 22\right)\cdot 23 + \left(4 a + 1\right)\cdot 23^{2} + \left(a + 12\right)\cdot 23^{3} + 11\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $4 a + 4 + \left(2 a + 22\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(21 a + 9\right)\cdot 23^{3} + \left(22 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 a + 3 + \left(12 a + 3\right)\cdot 23 + \left(7 a + 5\right)\cdot 23^{2} + \left(12 a + 10\right)\cdot 23^{3} + 6 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $7 a + 17 + \left(10 a + 6\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(13 a + 9\right)\cdot 23^{3} + \left(6 a + 11\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 6 }$ $=$ $16 a + 8 + \left(12 a + 20\right)\cdot 23 + \left(10 a + 15\right)\cdot 23^{2} + 9 a\cdot 23^{3} + \left(16 a + 11\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,3)(5,6)$ $-1$ $1$ $3$ $(1,5,3)(2,4,6)$ $\zeta_{3}$ $1$ $3$ $(1,3,5)(2,6,4)$ $-\zeta_{3} - 1$ $1$ $6$ $(1,2,5,4,3,6)$ $\zeta_{3} + 1$ $1$ $6$ $(1,6,3,4,5,2)$ $-\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.