Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 383 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 224\cdot 383 + 145\cdot 383^{2} + 361\cdot 383^{3} + 318\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 79 + 153\cdot 383 + 329\cdot 383^{2} + 47\cdot 383^{3} + 323\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 85 + 218\cdot 383 + 259\cdot 383^{2} + 299\cdot 383^{3} + 325\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 204 + 179\cdot 383 + 81\cdot 383^{2} + 380\cdot 383^{3} + 158\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 223 + 189\cdot 383 + 39\cdot 383^{2} + 361\cdot 383^{3} + 172\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 256 + 178\cdot 383 + 2\cdot 383^{2} + 108\cdot 383^{3} + 108\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 313 + 280\cdot 383 + 329\cdot 383^{2} + 355\cdot 383^{3} + 251\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 335 + 107\cdot 383 + 344\cdot 383^{2} + 255\cdot 383^{4} +O\left(383^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,2,4,8,6,7,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,7)(2,8)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
