# Properties

 Label 1.3e2_13.6t1.8c2 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 $f= x^{6} - 26 x^{3} + 351 x^{2} - 936 x + 793$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{117}(23,\cdot)$$

## 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:
 $r_{ 1 }$ $=$ $13 a + 9 + \left(a + 14\right)\cdot 23 + \left(22 a + 4\right)\cdot 23^{2} + \left(a + 4\right)\cdot 23^{3} + \left(3 a + 21\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 a + 12 + \left(9 a + 1\right)\cdot 23 + \left(3 a + 18\right)\cdot 23^{2} + \left(10 a + 18\right)\cdot 23^{3} + \left(14 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $13 a + 2 + \left(11 a + 7\right)\cdot 23 + 20 a\cdot 23^{2} + 10 a\cdot 23^{3} + \left(5 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $10 a + 5 + \left(11 a + 17\right)\cdot 23 + \left(2 a + 6\right)\cdot 23^{2} + \left(12 a + 1\right)\cdot 23^{3} + \left(17 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $3 a + 6 + \left(13 a + 1\right)\cdot 23 + \left(19 a + 15\right)\cdot 23^{2} + \left(12 a + 12\right)\cdot 23^{3} + \left(8 a + 8\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 6 }$ $=$ $10 a + 12 + \left(21 a + 4\right)\cdot 23 + 23^{2} + \left(21 a + 9\right)\cdot 23^{3} + \left(19 a + 2\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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