Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 251 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 32\cdot 251 + 89\cdot 251^{2} + 123\cdot 251^{3} + 56\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 + 68\cdot 251 + 139\cdot 251^{2} + 112\cdot 251^{3} + 146\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 69 + 51\cdot 251 + 18\cdot 251^{2} + 5\cdot 251^{3} + 31\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 88 + 17\cdot 251 + 109\cdot 251^{2} + 52\cdot 251^{3} + 31\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 102 + 119\cdot 251 + 225\cdot 251^{2} + 179\cdot 251^{3} + 101\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 108 + 222\cdot 251 + 149\cdot 251^{2} + 98\cdot 251^{3} + 129\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 152 + 39\cdot 251 + 190\cdot 251^{2} + 30\cdot 251^{3} + 87\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 165 + 202\cdot 251 + 82\cdot 251^{2} + 150\cdot 251^{3} + 169\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,7)(6,8)$ |
| $(1,6,8,5)(2,3,7,4)$ |
| $(1,4,8,3)(2,6,7,5)$ |
| $(1,8)(2,7)(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,4)(5,6)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,5)(2,7)(6,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,6,8,5)(2,3,7,4)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,3)(2,6,7,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,6,2,8,3,5,7)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,3,6,7,8,4,5,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
