Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 116\cdot 181 + 7\cdot 181^{2} + 85\cdot 181^{3} + 73\cdot 181^{4} + 73\cdot 181^{5} + 94\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 + 181 + 103\cdot 181^{2} + 166\cdot 181^{3} + 39\cdot 181^{4} + 95\cdot 181^{5} + 26\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 + 11\cdot 181 + 30\cdot 181^{2} + 74\cdot 181^{3} + 6\cdot 181^{4} + 135\cdot 181^{5} + 22\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 169\cdot 181 + 172\cdot 181^{2} + 86\cdot 181^{3} + 56\cdot 181^{4} + 91\cdot 181^{5} + 123\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 105 + 173\cdot 181 + 69\cdot 181^{2} + 46\cdot 181^{3} + 154\cdot 181^{4} + 165\cdot 181^{5} + 19\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 149 + 7\cdot 181 + 89\cdot 181^{2} + 154\cdot 181^{3} + 144\cdot 181^{4} + 150\cdot 181^{5} + 14\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 151 + 45\cdot 181 + 54\cdot 181^{2} + 48\cdot 181^{3} + 137\cdot 181^{4} + 2\cdot 181^{5} + 49\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 153 + 17\cdot 181 + 16\cdot 181^{2} + 62\cdot 181^{3} + 111\cdot 181^{4} + 9\cdot 181^{5} + 11\cdot 181^{6} +O\left(181^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,7,4)(2,3,8,6)$ |
| $(2,8)(4,5)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
| $(1,7)(2,8)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
| $(1,5,7,4)(2,6,8,3)$ |
| $(2,8)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(2,8)(3,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(3,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,4)(5,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,8)(4,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,7,4)(2,3,8,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,7,2)(3,5,6,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,7,3)(2,5,8,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,7,4)(2,6,8,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,7,8)(3,5,6,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,7,3)(2,4,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
