Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 52 + 73\cdot 181 + 23\cdot 181^{2} + 118\cdot 181^{3} + 98\cdot 181^{4} + 43\cdot 181^{5} + 33\cdot 181^{6} + 101\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 + 2\cdot 181 + 113\cdot 181^{2} + 177\cdot 181^{3} + 170\cdot 181^{4} + 158\cdot 181^{5} + 102\cdot 181^{6} + 105\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 69 + 95\cdot 181 + 116\cdot 181^{2} + 167\cdot 181^{3} + 77\cdot 181^{4} + 124\cdot 181^{5} + 95\cdot 181^{6} + 40\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 + 178\cdot 181 + 45\cdot 181^{2} + 93\cdot 181^{3} + 127\cdot 181^{4} + 93\cdot 181^{5} + 25\cdot 181^{6} + 98\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 90 + 174\cdot 181 + 47\cdot 181^{2} + 179\cdot 181^{3} + 63\cdot 181^{4} + 35\cdot 181^{5} + 86\cdot 181^{6} + 83\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 121 + 72\cdot 181 + 135\cdot 181^{2} + 117\cdot 181^{3} + 96\cdot 181^{4} + 76\cdot 181^{5} + 77\cdot 181^{6} + 26\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 124 + 94\cdot 181 + 151\cdot 181^{2} + 102\cdot 181^{3} + 92\cdot 181^{4} + 108\cdot 181^{5} + 154\cdot 181^{6} + 139\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 132 + 32\cdot 181 + 90\cdot 181^{2} + 129\cdot 181^{3} + 176\cdot 181^{4} + 82\cdot 181^{5} + 148\cdot 181^{6} + 128\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(6,8)$ |
| $(1,7,8,3,2,4,6,5)$ |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,8,2,6)(3,4,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,7)(6,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(6,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,8,2,6)(3,4,5,7)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,6,2,8)(3,7,5,4)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,6,2,8)(3,4,5,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,8,3,2,4,6,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,6,7,2,5,8,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,6,3,2,7,8,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,8,4,2,5,6,7)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
