# Properties

 Label 1.1001.6t1.a Dimension 1 Group $C_6$ Conductor $7 \cdot 11 \cdot 13$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $1001= 7 \cdot 11 \cdot 13$ Artin number field: Splitting field of $f= x^{6} - x^{5} - 52 x^{4} + 109 x^{3} + 897 x^{2} - 3168 x + 4887$ 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:
 $r_{ 1 }$ $=$ $8 a + 5 + \left(9 a + 5\right)\cdot 19 + 8 a\cdot 19^{2} + \left(3 a + 5\right)\cdot 19^{3} + \left(4 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 a + 2 + \left(9 a + 16\right)\cdot 19 + \left(10 a + 9\right)\cdot 19^{2} + \left(15 a + 3\right)\cdot 19^{3} + \left(14 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 a + 7 + \left(9 a + 17\right)\cdot 19 + \left(10 a + 17\right)\cdot 19^{2} + \left(15 a + 7\right)\cdot 19^{3} + \left(14 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 a + 13 + \left(9 a + 6\right)\cdot 19 + \left(10 a + 18\right)\cdot 19^{2} + \left(15 a + 18\right)\cdot 19^{3} + \left(14 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $8 a + 13 + \left(9 a + 14\right)\cdot 19 + \left(8 a + 10\right)\cdot 19^{2} + \left(3 a + 8\right)\cdot 19^{3} + \left(4 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 6 }$ $=$ $8 a + 18 + \left(9 a + 15\right)\cdot 19 + \left(8 a + 18\right)\cdot 19^{2} + \left(3 a + 12\right)\cdot 19^{3} + \left(4 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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