# Properties

 Label 1.3_7.6t1.1 Dimension 1 Group $C_6$ Conductor $3 \cdot 7$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $21= 3 \cdot 7$ Artin number field: Splitting field of $f= x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Even

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,3)(2,6)(4,5)$ $-1$ $-1$ $1$ $3$ $(1,4,6)(2,3,5)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,6,4)(2,5,3)$ $-\zeta_{3} - 1$ $\zeta_{3}$ $1$ $6$ $(1,2,4,3,6,5)$ $\zeta_{3} + 1$ $-\zeta_{3}$ $1$ $6$ $(1,5,6,3,4,2)$ $-\zeta_{3}$ $\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.