# Properties

 Label 1.5_11.10t1.1 Dimension 1 Group $C_{10}$ Conductor $5 \cdot 11$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $55= 5 \cdot 11$ Artin number field: Splitting field of $f= x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_{10}$ Parity: Even

## 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 $\mathbb{Q}_{ 19 }$: $x^{5} + 5 x + 17$
Roots:
 $r_{ 1 }$ $=$ $2 a^{4} + 11 a^{3} + 8 a^{2} + 17 a + 9 + \left(a^{4} + 14 a^{3} + 11 a + 12\right)\cdot 19 + \left(7 a^{4} + 6 a^{3} + 2 a + 3\right)\cdot 19^{2} + \left(16 a^{4} + 4 a^{3} + 6 a^{2} + 18 a + 15\right)\cdot 19^{3} + \left(5 a^{4} + 9 a^{3} + 3 a^{2} + 15 a + 9\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $2 a^{4} + 14 a^{3} + 2 a^{2} + 17 a + 11 + \left(5 a^{4} + 3 a^{3} + 11 a^{2} + 16 a + 8\right)\cdot 19 + \left(a^{4} + a^{3} + 13 a^{2} + 12 a + 14\right)\cdot 19^{2} + \left(4 a^{4} + 6 a^{3} + 4 a^{2} + 15 a + 5\right)\cdot 19^{3} + \left(8 a^{4} + 8 a^{3} + 8 a^{2} + 16 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $4 a^{4} + 15 a^{3} + 17 a^{2} + 4 a + 17 + \left(6 a^{4} + 11 a^{3} + 15 a^{2} + 2 a + 13\right)\cdot 19 + \left(6 a^{4} + 14 a^{3} + 9 a^{2} + 5 a\right)\cdot 19^{2} + \left(8 a^{4} + a^{3} + 3 a^{2} + 4 a + 2\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 12 a^{2} + 3 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $5 a^{4} + a^{3} + 2 a^{2} + 4 + \left(16 a^{4} + a^{3} + 16 a^{2} + 10 a + 15\right)\cdot 19 + \left(2 a^{4} + 3 a^{3} + 8 a^{2} + 1\right)\cdot 19^{2} + \left(18 a^{4} + 15 a^{3} + 5 a^{2} + 10 a + 5\right)\cdot 19^{3} + \left(2 a^{4} + 9 a^{3} + 16 a^{2} + 17 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $6 a^{4} + 14 a^{3} + 5 a^{2} + 13 a + 8 + \left(7 a^{4} + 8 a^{3} + 17 a^{2} + 12 a + 17\right)\cdot 19 + \left(5 a^{4} + 7 a^{3} + 4 a^{2} + 11 a + 11\right)\cdot 19^{2} + \left(7 a^{4} + 3 a^{3} + a^{2} + 17 a + 18\right)\cdot 19^{3} + \left(17 a^{4} + 5 a^{3} + 8 a^{2} + 15 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 6 }$ $=$ $7 a^{4} + 11 a^{3} + 7 a^{2} + 12 a + 10 + \left(3 a^{4} + 6 a^{3} + 5 a^{2} + 16 a + 2\right)\cdot 19 + \left(16 a^{3} + 9 a^{2} + 3 a + 14\right)\cdot 19^{2} + \left(8 a^{4} + 12 a^{3} + 14 a^{2} + 5 a\right)\cdot 19^{3} + \left(15 a^{4} + 3 a^{3} + 15 a^{2} + 10 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 7 }$ $=$ $8 a^{4} + 13 a^{3} + 7 a^{2} + 14 + \left(6 a^{4} + 10 a^{3} + 13 a^{2} + 16 a + 14\right)\cdot 19 + \left(14 a^{4} + 7 a^{3} + 8 a^{2} + 14 a + 13\right)\cdot 19^{2} + \left(13 a^{4} + 11 a^{3} + 7 a^{2} + 9 a + 4\right)\cdot 19^{3} + \left(15 a^{4} + 10 a^{3} + 14 a + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 8 }$ $=$ $12 a^{4} + 7 a^{3} + 13 a^{2} + 12 a + 13 + \left(7 a^{4} + 8 a^{3} + 5 a^{2} + 14 a + 18\right)\cdot 19 + \left(14 a^{4} + 8 a^{3} + 3 a^{2} + 2 a + 9\right)\cdot 19^{2} + \left(3 a^{4} + a^{3} + 9 a^{2} + 6 a + 4\right)\cdot 19^{3} + \left(12 a^{4} + 18 a^{3} + 8 a^{2} + 14 a + 9\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 9 }$ $=$ $13 a^{4} + 2 a^{3} + 16 a^{2} + 15 a + 17 + \left(a^{4} + 16 a^{3} + 6 a^{2} + 2 a + 13\right)\cdot 19 + \left(14 a^{4} + 17 a^{3} + 7 a^{2} + 10 a + 8\right)\cdot 19^{2} + \left(4 a^{4} + 11 a^{3} + 17 a^{2} + 7 a + 8\right)\cdot 19^{3} + \left(16 a^{4} + 15 a^{3} + 15 a^{2} + 11 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 10 }$ $=$ $17 a^{4} + 7 a^{3} + 18 a^{2} + 5 a + 12 + \left(a^{4} + 13 a^{3} + 2 a^{2} + 10 a + 15\right)\cdot 19 + \left(10 a^{4} + 11 a^{3} + 10 a^{2} + 11 a + 15\right)\cdot 19^{2} + \left(10 a^{4} + 7 a^{3} + 6 a^{2} + 10\right)\cdot 19^{3} + \left(17 a^{4} + 15 a^{3} + 6 a^{2} + 13 a + 18\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

 Cycle notation $(1,8,6,9,7,5,3,2,10,4)$ $(1,5)(2,6)(3,8)(4,7)(9,10)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character values $c1$ $c2$ $c3$ $c4$ $1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $2$ $(1,5)(2,6)(3,8)(4,7)(9,10)$ $-1$ $-1$ $-1$ $-1$ $1$ $5$ $(1,6,7,3,10)(2,4,8,9,5)$ $\zeta_{5}$ $\zeta_{5}^{2}$ $\zeta_{5}^{3}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $1$ $5$ $(1,7,10,6,3)(2,8,5,4,9)$ $\zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}$ $\zeta_{5}^{3}$ $1$ $5$ $(1,3,6,10,7)(2,9,4,5,8)$ $\zeta_{5}^{3}$ $\zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{2}$ $1$ $5$ $(1,10,3,7,6)(2,5,9,8,4)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{3}$ $\zeta_{5}^{2}$ $\zeta_{5}$ $1$ $10$ $(1,8,6,9,7,5,3,2,10,4)$ $-\zeta_{5}^{3}$ $-\zeta_{5}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $-\zeta_{5}^{2}$ $1$ $10$ $(1,9,3,4,6,5,10,8,7,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $-\zeta_{5}^{3}$ $-\zeta_{5}^{2}$ $-\zeta_{5}$ $1$ $10$ $(1,2,7,8,10,5,6,4,3,9)$ $-\zeta_{5}$ $-\zeta_{5}^{2}$ $-\zeta_{5}^{3}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $1$ $10$ $(1,4,10,2,3,5,7,9,6,8)$ $-\zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $-\zeta_{5}$ $-\zeta_{5}^{3}$
The blue line marks the conjugacy class containing complex conjugation.