# Properties

 Label 2.2880.8t17.a Dimension 2 Group $C_4\wr C_2$ Conductor $2^{6} \cdot 3^{2} \cdot 5$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $2880= 2^{6} \cdot 3^{2} \cdot 5$ Artin number field: Splitting field of $f= x^{8} - 6 x^{4} + 45$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4\wr C_2$ Parity: Odd

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 11.
Roots:
 $r_{ 1 }$ $=$ $2 + 3\cdot 41 + 6\cdot 41^{2} + 33\cdot 41^{3} + 33\cdot 41^{4} + 20\cdot 41^{5} + 24\cdot 41^{6} + 34\cdot 41^{7} + 15\cdot 41^{8} + 26\cdot 41^{9} + 18\cdot 41^{10} +O\left(41^{ 11 }\right)$ $r_{ 2 }$ $=$ $12 + 33\cdot 41 + 33\cdot 41^{2} + 29\cdot 41^{3} + 19\cdot 41^{4} + 17\cdot 41^{5} + 27\cdot 41^{6} + 37\cdot 41^{7} + 10\cdot 41^{8} + 19\cdot 41^{9} + 31\cdot 41^{10} +O\left(41^{ 11 }\right)$ $r_{ 3 }$ $=$ $15 + 2\cdot 41 + 13\cdot 41^{2} + 40\cdot 41^{3} + 7\cdot 41^{4} + 25\cdot 41^{5} + 24\cdot 41^{6} + 8\cdot 41^{7} + 21\cdot 41^{8} + 14\cdot 41^{9} + 31\cdot 41^{10} +O\left(41^{ 11 }\right)$ $r_{ 4 }$ $=$ $18 + 4\cdot 41 + 27\cdot 41^{2} + 8\cdot 41^{3} + 22\cdot 41^{4} + 20\cdot 41^{5} + 34\cdot 41^{6} + 8\cdot 41^{7} + 33\cdot 41^{8} + 3\cdot 41^{9} +O\left(41^{ 11 }\right)$ $r_{ 5 }$ $=$ $23 + 36\cdot 41 + 13\cdot 41^{2} + 32\cdot 41^{3} + 18\cdot 41^{4} + 20\cdot 41^{5} + 6\cdot 41^{6} + 32\cdot 41^{7} + 7\cdot 41^{8} + 37\cdot 41^{9} + 40\cdot 41^{10} +O\left(41^{ 11 }\right)$ $r_{ 6 }$ $=$ $26 + 38\cdot 41 + 27\cdot 41^{2} + 33\cdot 41^{4} + 15\cdot 41^{5} + 16\cdot 41^{6} + 32\cdot 41^{7} + 19\cdot 41^{8} + 26\cdot 41^{9} + 9\cdot 41^{10} +O\left(41^{ 11 }\right)$ $r_{ 7 }$ $=$ $29 + 7\cdot 41 + 7\cdot 41^{2} + 11\cdot 41^{3} + 21\cdot 41^{4} + 23\cdot 41^{5} + 13\cdot 41^{6} + 3\cdot 41^{7} + 30\cdot 41^{8} + 21\cdot 41^{9} + 9\cdot 41^{10} +O\left(41^{ 11 }\right)$ $r_{ 8 }$ $=$ $39 + 37\cdot 41 + 34\cdot 41^{2} + 7\cdot 41^{3} + 7\cdot 41^{4} + 20\cdot 41^{5} + 16\cdot 41^{6} + 6\cdot 41^{7} + 25\cdot 41^{8} + 14\cdot 41^{9} + 22\cdot 41^{10} +O\left(41^{ 11 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $-2$ $2$ $2$ $(1,8)(4,5)$ $0$ $0$ $4$ $2$ $(1,6)(2,4)(3,8)(5,7)$ $0$ $0$ $1$ $4$ $(1,5,8,4)(2,6,7,3)$ $-2 \zeta_{4}$ $2 \zeta_{4}$ $1$ $4$ $(1,4,8,5)(2,3,7,6)$ $2 \zeta_{4}$ $-2 \zeta_{4}$ $2$ $4$ $(1,5,8,4)(2,3,7,6)$ $0$ $0$ $2$ $4$ $(1,5,8,4)$ $-\zeta_{4} + 1$ $\zeta_{4} + 1$ $2$ $4$ $(1,4,8,5)$ $\zeta_{4} + 1$ $-\zeta_{4} + 1$ $2$ $4$ $(1,8)(2,3,7,6)(4,5)$ $\zeta_{4} - 1$ $-\zeta_{4} - 1$ $2$ $4$ $(1,8)(2,6,7,3)(4,5)$ $-\zeta_{4} - 1$ $\zeta_{4} - 1$ $4$ $4$ $(1,6,8,3)(2,5,7,4)$ $0$ $0$ $4$ $8$ $(1,6,5,7,8,3,4,2)$ $0$ $0$ $4$ $8$ $(1,7,4,6,8,2,5,3)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.