# Properties

 Label 1.113.8t1.1 Dimension 1 Group $C_8$ Conductor $113$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $113$ Artin number field: Splitting field of $f= x^{8} - x^{7} - 49 x^{6} - 16 x^{5} + 511 x^{4} + 367 x^{3} - 1499 x^{2} - 798 x + 1372$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_8$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 80\cdot 97 + 37\cdot 97^{2} + 72\cdot 97^{3} + 43\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 2 }$ $=$ $10 + 34\cdot 97 + 60\cdot 97^{2} + 67\cdot 97^{3} + 2\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 3 }$ $=$ $27 + 82\cdot 97 + 89\cdot 97^{2} + 44\cdot 97^{3} + 13\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 4 }$ $=$ $30 + 40\cdot 97 + 44\cdot 97^{2} + 30\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 5 }$ $=$ $31 + 39\cdot 97 + 24\cdot 97^{2} + 23\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 6 }$ $=$ $55 + 42\cdot 97 + 38\cdot 97^{2} + 11\cdot 97^{3} + 86\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 7 }$ $=$ $59 + 74\cdot 97 + 29\cdot 97^{2} + 84\cdot 97^{3} + 38\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 8 }$ $=$ $77 + 91\cdot 97 + 62\cdot 97^{2} + 83\cdot 97^{3} + 96\cdot 97^{4} +O\left(97^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

### Character values on conjugacy classes

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