# Properties

 Label 1.1035.6t1.a Dimension 1 Group $C_6$ Conductor $3^{2} \cdot 5 \cdot 23$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $1035= 3^{2} \cdot 5 \cdot 23$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 84 x^{4} - 161 x^{3} + 2607 x^{2} - 2706 x + 29699$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $7 a + 6 + \left(14 a + 14\right)\cdot 19 + \left(4 a + 15\right)\cdot 19^{2} + \left(3 a + 12\right)\cdot 19^{3} + \left(13 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $7 a + \left(14 a + 2\right)\cdot 19 + \left(4 a + 9\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(13 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 a + 7 + \left(4 a + 9\right)\cdot 19 + \left(14 a + 18\right)\cdot 19^{2} + \left(15 a + 7\right)\cdot 19^{3} + \left(5 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $12 a + 11 + \left(4 a + 8\right)\cdot 19 + \left(14 a + 8\right)\cdot 19^{2} + \left(15 a + 16\right)\cdot 19^{3} + \left(5 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $12 a + 13 + \left(4 a + 2\right)\cdot 19 + \left(14 a + 6\right)\cdot 19^{2} + \left(15 a + 11\right)\cdot 19^{3} + \left(5 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 6 }$ $=$ $7 a + 4 + \left(14 a + 1\right)\cdot 19 + \left(4 a + 18\right)\cdot 19^{2} + \left(3 a + 17\right)\cdot 19^{3} + \left(13 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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