Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 2\cdot 181 + 87\cdot 181^{2} + 99\cdot 181^{3} + 14\cdot 181^{4} + 4\cdot 181^{5} + 154\cdot 181^{6} + 101\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 + 110\cdot 181 + 153\cdot 181^{2} + 170\cdot 181^{3} + 130\cdot 181^{4} + 35\cdot 181^{5} + 142\cdot 181^{6} + 134\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 41\cdot 181 + 54\cdot 181^{2} + 173\cdot 181^{3} + 83\cdot 181^{4} + 45\cdot 181^{5} + 121\cdot 181^{6} + 31\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 178\cdot 181 + 85\cdot 181^{2} + 125\cdot 181^{3} + 95\cdot 181^{4} + 112\cdot 181^{5} + 150\cdot 181^{6} + 170\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 111 + 74\cdot 181 + 147\cdot 181^{2} + 45\cdot 181^{3} + 166\cdot 181^{4} + 96\cdot 181^{5} + 125\cdot 181^{6} + 140\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 125 + 36\cdot 181 + 46\cdot 181^{2} + 87\cdot 181^{3} + 121\cdot 181^{4} + 68\cdot 181^{5} + 78\cdot 181^{6} + 153\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 163 + 143\cdot 181 + 51\cdot 181^{2} + 71\cdot 181^{3} + 181^{4} + 164\cdot 181^{5} + 86\cdot 181^{6} + 32\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 174 + 136\cdot 181 + 97\cdot 181^{2} + 131\cdot 181^{3} + 109\cdot 181^{4} + 15\cdot 181^{5} + 46\cdot 181^{6} + 139\cdot 181^{7} +O\left(181^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,8)(3,4)(6,7)$ |
| $(3,4)(6,7)$ |
| $(1,8,5,2)(3,7,4,6)$ |
| $(3,6,4,7)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,8)(3,4)(6,7)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,4)(6,7)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,6)(4,5)(7,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,8,5,2)(3,7,4,6)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,5,8)(3,6,4,7)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(3,6,4,7)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(3,7,4,6)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,5)(2,8)(3,7,4,6)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,5)(2,8)(3,6,4,7)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,8,5,2)(3,6,4,7)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,4,5,3)(2,7,8,6)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,6,2,4,5,7,8,3)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,4,8,6,5,3,2,7)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
