# Properties

 Label 1.5_19.6t1.1 Dimension 1 Group $C_6$ Conductor $5 \cdot 19$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $95= 5 \cdot 19$ Artin number field: Splitting field of $f= x^{6} - x^{5} - 16 x^{4} + x^{3} + 47 x^{2} + 10 x - 11$ 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 $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$
Roots:
 $r_{ 1 }$ $=$ $25 a + 35 + \left(8 a + 13\right)\cdot 37 + \left(16 a + 36\right)\cdot 37^{2} + \left(29 a + 13\right)\cdot 37^{3} + \left(13 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 a + 25 + \left(28 a + 16\right)\cdot 37 + \left(20 a + 32\right)\cdot 37^{2} + \left(7 a + 22\right)\cdot 37^{3} + \left(23 a + 8\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 a + 24 + \left(28 a + 23\right)\cdot 37 + \left(20 a + 18\right)\cdot 37^{2} + \left(7 a + 4\right)\cdot 37^{3} + \left(23 a + 33\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $25 a + 36 + \left(8 a + 6\right)\cdot 37 + \left(16 a + 13\right)\cdot 37^{2} + \left(29 a + 32\right)\cdot 37^{3} + \left(13 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 5 }$ $=$ $12 a + 9 + \left(28 a + 11\right)\cdot 37 + \left(20 a + 33\right)\cdot 37^{2} + \left(7 a + 13\right)\cdot 37^{3} + \left(23 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 6 }$ $=$ $25 a + 20 + \left(8 a + 1\right)\cdot 37 + \left(16 a + 14\right)\cdot 37^{2} + \left(29 a + 23\right)\cdot 37^{3} + \left(13 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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