# Properties

 Label 1.31.10t1.1c1 Dimension 1 Group $C_{10}$ Conductor $31$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $31$ Artin number field: Splitting field of $f= x^{10} - x^{9} + 2 x^{8} + 16 x^{7} - 9 x^{6} + 11 x^{5} + 43 x^{4} - 6 x^{3} + 63 x^{2} - 20 x + 25$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_{10}$ Parity: Odd Corresponding Dirichlet character: $$\chi_{31}(29,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{5} + 5 x + 17$
Roots:
 $r_{ 1 }$ $=$ $a^{3} + 8 a^{2} + 16 a + 18 + \left(12 a^{4} + 6 a^{3} + 12 a^{2} + 17 a + 8\right)\cdot 19 + \left(7 a^{4} + 5 a^{3} + 7 a^{2} + 18 a + 16\right)\cdot 19^{2} + \left(7 a^{4} + 2 a^{3} + 7 a^{2} + 16 a + 18\right)\cdot 19^{3} + \left(18 a^{4} + 2 a^{3} + 6 a^{2} + 2 a + 17\right)\cdot 19^{4} + \left(a^{4} + 13 a^{3} + 17 a^{2} + 9 a + 7\right)\cdot 19^{5} + \left(11 a^{4} + 18 a^{3} + 17 a^{2} + 13 a + 8\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 2 }$ $=$ $a^{4} + 14 a^{3} + 4 a + 3 + \left(15 a^{4} + 14 a^{3} + 6 a^{2} + 12 a + 2\right)\cdot 19 + \left(17 a^{4} + 14 a^{3} + 8 a^{2} + 2 a\right)\cdot 19^{2} + \left(7 a^{4} + 14 a^{3} + 11 a^{2} + 8 a + 2\right)\cdot 19^{3} + \left(18 a^{4} + 7 a^{3} + 15 a^{2} + 11 a + 18\right)\cdot 19^{4} + \left(12 a^{3} + 5 a^{2} + 18 a + 3\right)\cdot 19^{5} + \left(9 a^{4} + 2 a^{3} + 17 a^{2} + 4 a\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 3 }$ $=$ $a^{4} + 18 a^{3} + 11 a^{2} + 16 a + 9 + \left(3 a^{4} + 2 a^{3} + 18 a^{2} + 2 a + 9\right)\cdot 19 + \left(18 a^{4} + 12 a^{3} + 7 a^{2} + 5 a + 14\right)\cdot 19^{2} + \left(11 a^{4} + 3 a^{3} + 9 a^{2} + 17 a + 16\right)\cdot 19^{3} + \left(15 a^{4} + 15 a^{3} + 12 a^{2} + 18 a + 7\right)\cdot 19^{4} + \left(5 a^{4} + 8 a^{3} + 14 a^{2} + 14 a\right)\cdot 19^{5} + \left(16 a^{3} + 9 a^{2} + 6 a + 3\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 4 }$ $=$ $3 a^{4} + 12 a^{3} + a^{2} + 16 a + 11 + \left(18 a^{4} + 5 a^{3} + 17 a^{2} + 13 a + 14\right)\cdot 19 + \left(10 a^{4} + 7 a^{3} + 5 a^{2} + 18 a + 10\right)\cdot 19^{2} + \left(14 a^{4} + 18 a^{3} + 10 a^{2} + 4 a + 9\right)\cdot 19^{3} + \left(16 a^{3} + 13 a^{2} + 2 a + 4\right)\cdot 19^{4} + \left(12 a^{4} + 13 a^{3} + 18 a^{2} + 4 a + 10\right)\cdot 19^{5} + \left(11 a^{4} + 8 a^{3} + 11 a^{2} + 17 a + 10\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 5 }$ $=$ $10 a^{4} + 6 a^{3} + 7 a^{2} + 12 a + 7 + \left(4 a^{4} + 7 a^{3} + 14 a^{2} + 3 a + 15\right)\cdot 19 + \left(12 a^{4} + 11 a^{3} + 18 a^{2} + 11 a + 9\right)\cdot 19^{2} + \left(14 a^{4} + 9 a^{3} + 12 a^{2} + 16 a + 8\right)\cdot 19^{3} + \left(12 a^{4} + 18 a^{3} + 2 a^{2} + 15\right)\cdot 19^{4} + \left(6 a^{4} + 18 a^{3} + 3 a^{2} + 8 a + 3\right)\cdot 19^{5} + \left(2 a^{3} + 9 a^{2} + 6 a + 3\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 6 }$ $=$ $11 a^{4} + 17 a^{3} + a^{2} + 17 a + 11 + \left(4 a^{4} + 13 a^{3} + 12 a^{2} + 4 a + 15\right)\cdot 19 + \left(15 a^{4} + 15 a^{3} + a^{2} + a + 2\right)\cdot 19^{2} + \left(17 a^{4} + 2 a^{3} + 18 a^{2} + 12 a + 2\right)\cdot 19^{3} + \left(12 a^{4} + 13 a^{3} + 8 a^{2} + 8 a + 16\right)\cdot 19^{4} + \left(a^{4} + 8 a^{3} + 16 a^{2} + 4 a + 2\right)\cdot 19^{5} + \left(13 a^{4} + 14 a^{3} + 16 a^{2} + 3 a + 16\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 7 }$ $=$ $16 a^{4} + 12 a^{3} + 17 a^{2} + 11 a + 6 + \left(16 a^{4} + 16 a^{3} + 16 a^{2} + 14 a + 9\right)\cdot 19 + \left(16 a^{4} + 2 a^{3} + 5 a^{2} + 14 a + 15\right)\cdot 19^{2} + \left(10 a^{4} + 16 a^{3} + 6 a^{2} + 17 a + 13\right)\cdot 19^{3} + \left(4 a^{4} + 5 a^{3} + a^{2} + a\right)\cdot 19^{4} + \left(10 a^{4} + 10 a^{3} + 11 a^{2} + 2 a + 3\right)\cdot 19^{5} + \left(12 a^{4} + 12 a^{3} + 7 a^{2} + 14\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 8 }$ $=$ $17 a^{4} + 17 a^{3} + 13 a^{2} + 16 a + 16 + \left(4 a^{4} + 13 a^{3} + 9 a^{2} + 2 a + 16\right)\cdot 19 + \left(14 a^{4} + 9 a^{3} + 2 a^{2} + 5 a + 17\right)\cdot 19^{2} + \left(a^{4} + 4 a^{3} + 18 a^{2} + 13\right)\cdot 19^{3} + \left(11 a^{4} + 5 a^{3} + 15 a^{2} + 12 a + 8\right)\cdot 19^{4} + \left(8 a^{4} + 5 a^{3} + 13 a^{2} + 7 a + 11\right)\cdot 19^{5} + \left(4 a^{4} + a^{3} + 9 a^{2} + 2 a\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 9 }$ $=$ $18 a^{4} + 18 a^{3} + 6 a^{2} + 15 a + 1 + \left(a^{4} + 18 a^{3} + 2 a^{2} + 4 a + 5\right)\cdot 19 + \left(16 a^{4} + 7 a^{3} + 7 a^{2} + 15 a + 6\right)\cdot 19^{2} + \left(10 a^{4} + 17 a^{3} + 17 a^{2} + 10 a + 12\right)\cdot 19^{3} + \left(4 a^{4} + 4 a^{3} + 16 a^{2} + 16 a + 1\right)\cdot 19^{4} + \left(15 a^{4} + 15 a^{3} + 8 a^{2} + 2 a\right)\cdot 19^{5} + \left(2 a^{3} + 11 a^{2} + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ $r_{ 10 }$ $=$ $18 a^{4} + 18 a^{3} + 12 a^{2} + 10 a + 14 + \left(13 a^{4} + 13 a^{3} + 4 a^{2} + 17 a + 16\right)\cdot 19 + \left(3 a^{4} + 7 a^{3} + 10 a^{2} + a\right)\cdot 19^{2} + \left(16 a^{4} + 5 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 19^{3} + \left(14 a^{4} + 5 a^{3} + a^{2} + 3\right)\cdot 19^{4} + \left(12 a^{4} + 7 a^{3} + 4 a^{2} + 4 a + 13\right)\cdot 19^{5} + \left(12 a^{4} + 14 a^{3} + 2 a^{2} + 2 a + 14\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$

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

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

### Character values on conjugacy classes

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