Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 13.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 + 66\cdot 181 + 171\cdot 181^{2} + 10\cdot 181^{3} + 68\cdot 181^{4} + 173\cdot 181^{5} + 123\cdot 181^{6} + 6\cdot 181^{7} + 127\cdot 181^{8} + 18\cdot 181^{9} + 102\cdot 181^{10} + 5\cdot 181^{11} + 167\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 + 91\cdot 181 + 23\cdot 181^{2} + 9\cdot 181^{3} + 4\cdot 181^{4} + 140\cdot 181^{5} + 44\cdot 181^{6} + 17\cdot 181^{8} + 76\cdot 181^{9} + 51\cdot 181^{10} + 59\cdot 181^{11} + 166\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 67\cdot 181 + 168\cdot 181^{2} + 135\cdot 181^{3} + 104\cdot 181^{4} + 121\cdot 181^{5} + 68\cdot 181^{6} + 42\cdot 181^{7} + 140\cdot 181^{8} + 5\cdot 181^{9} + 135\cdot 181^{10} + 121\cdot 181^{11} + 114\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 84 + 68\cdot 181 + 91\cdot 181^{2} + 88\cdot 181^{3} + 11\cdot 181^{4} + 65\cdot 181^{5} + 122\cdot 181^{6} + 181^{7} + 154\cdot 181^{8} + 83\cdot 181^{9} + 145\cdot 181^{10} + 58\cdot 181^{11} + 172\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 97 + 112\cdot 181 + 89\cdot 181^{2} + 92\cdot 181^{3} + 169\cdot 181^{4} + 115\cdot 181^{5} + 58\cdot 181^{6} + 179\cdot 181^{7} + 26\cdot 181^{8} + 97\cdot 181^{9} + 35\cdot 181^{10} + 122\cdot 181^{11} + 8\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 137 + 113\cdot 181 + 12\cdot 181^{2} + 45\cdot 181^{3} + 76\cdot 181^{4} + 59\cdot 181^{5} + 112\cdot 181^{6} + 138\cdot 181^{7} + 40\cdot 181^{8} + 175\cdot 181^{9} + 45\cdot 181^{10} + 59\cdot 181^{11} + 66\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 144 + 89\cdot 181 + 157\cdot 181^{2} + 171\cdot 181^{3} + 176\cdot 181^{4} + 40\cdot 181^{5} + 136\cdot 181^{6} + 180\cdot 181^{7} + 163\cdot 181^{8} + 104\cdot 181^{9} + 129\cdot 181^{10} + 121\cdot 181^{11} + 14\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 158 + 114\cdot 181 + 9\cdot 181^{2} + 170\cdot 181^{3} + 112\cdot 181^{4} + 7\cdot 181^{5} + 57\cdot 181^{6} + 174\cdot 181^{7} + 53\cdot 181^{8} + 162\cdot 181^{9} + 78\cdot 181^{10} + 175\cdot 181^{11} + 13\cdot 181^{12} +O\left(181^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,8,2)(3,5,6,4)$ |
| $(2,7)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,5,7,6,8,4,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,7,6,8,4,2,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,2,5,8,3,7,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,7,6,8,5,2,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,2,4,8,3,7,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
