Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 251 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 52 + 146\cdot 251 + 94\cdot 251^{2} + 57\cdot 251^{3} + 99\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 78 + 46\cdot 251 + 156\cdot 251^{2} + 142\cdot 251^{3} + 247\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 88 + 155\cdot 251 + 204\cdot 251^{2} + 13\cdot 251^{3} + 42\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 95 + 99\cdot 251 + 163\cdot 251^{2} + 160\cdot 251^{3} + 36\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 104 + 228\cdot 251 + 117\cdot 251^{2} + 74\cdot 251^{3} + 120\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 135 + 6\cdot 251 + 88\cdot 251^{2} + 177\cdot 251^{3} + 79\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 212 + 100\cdot 251 + 130\cdot 251^{2} + 59\cdot 251^{3} + 43\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 241 + 220\cdot 251 + 48\cdot 251^{2} + 67\cdot 251^{3} + 84\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,7,6,8,5,4)$ |
| $(1,2,6,5)(3,7,8,4)$ |
| $(1,6)(2,5)(3,8)(4,7)$ |
| $(1,5)(2,6)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,6)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,2,6,5)(3,7,8,4)$ | $0$ |
| $4$ | $4$ | $(1,3,6,8)(2,4,5,7)$ | $0$ |
| $2$ | $8$ | $(1,3,2,7,6,8,5,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,2,4,6,3,5,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
