Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 104\cdot 191 + 138\cdot 191^{2} + 168\cdot 191^{3} + 51\cdot 191^{4} + 28\cdot 191^{5} + 65\cdot 191^{6} + 88\cdot 191^{7} + 47\cdot 191^{8} + 84\cdot 191^{9} + 39\cdot 191^{10} + 179\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 74\cdot 191 + 122\cdot 191^{2} + 87\cdot 191^{3} + 5\cdot 191^{4} + 58\cdot 191^{5} + 190\cdot 191^{6} + 17\cdot 191^{7} + 49\cdot 191^{8} + 100\cdot 191^{9} + 144\cdot 191^{10} + 38\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 118\cdot 191 + 16\cdot 191^{2} + 154\cdot 191^{3} + 174\cdot 191^{4} + 145\cdot 191^{5} + 95\cdot 191^{6} + 107\cdot 191^{7} + 161\cdot 191^{8} + 14\cdot 191^{9} + 69\cdot 191^{10} + 120\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 136\cdot 191 + 8\cdot 191^{2} + 54\cdot 191^{3} + 175\cdot 191^{4} + 128\cdot 191^{5} + 187\cdot 191^{6} + 133\cdot 191^{7} + 105\cdot 191^{8} + 6\cdot 191^{9} + 61\cdot 191^{10} + 36\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 151 + 54\cdot 191 + 182\cdot 191^{2} + 136\cdot 191^{3} + 15\cdot 191^{4} + 62\cdot 191^{5} + 3\cdot 191^{6} + 57\cdot 191^{7} + 85\cdot 191^{8} + 184\cdot 191^{9} + 129\cdot 191^{10} + 154\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 153 + 72\cdot 191 + 174\cdot 191^{2} + 36\cdot 191^{3} + 16\cdot 191^{4} + 45\cdot 191^{5} + 95\cdot 191^{6} + 83\cdot 191^{7} + 29\cdot 191^{8} + 176\cdot 191^{9} + 121\cdot 191^{10} + 70\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 172 + 116\cdot 191 + 68\cdot 191^{2} + 103\cdot 191^{3} + 185\cdot 191^{4} + 132\cdot 191^{5} + 173\cdot 191^{7} + 141\cdot 191^{8} + 90\cdot 191^{9} + 46\cdot 191^{10} + 152\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 188 + 86\cdot 191 + 52\cdot 191^{2} + 22\cdot 191^{3} + 139\cdot 191^{4} + 162\cdot 191^{5} + 125\cdot 191^{6} + 102\cdot 191^{7} + 143\cdot 191^{8} + 106\cdot 191^{9} + 151\cdot 191^{10} + 11\cdot 191^{11} +O\left(191^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,2,8,7)(3,5,6,4)$ |
| $(2,7)(3,4)(5,6)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
