Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 47 + 76\cdot 181 + 171\cdot 181^{2} + 98\cdot 181^{3} + 151\cdot 181^{4} + 162\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 + 50\cdot 181 + 118\cdot 181^{2} + 62\cdot 181^{3} + 25\cdot 181^{4} + 85\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 + 9\cdot 181 + 66\cdot 181^{2} + 156\cdot 181^{3} + 62\cdot 181^{4} + 61\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 86 + 177\cdot 181 + 11\cdot 181^{2} + 126\cdot 181^{3} + 83\cdot 181^{4} + 105\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 92 + 71\cdot 181 + 173\cdot 181^{2} + 86\cdot 181^{3} + 122\cdot 181^{4} + 3\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 99 + 65\cdot 181 + 6\cdot 181^{2} + 3\cdot 181^{3} + 166\cdot 181^{4} + 165\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 115 + 103\cdot 181 + 110\cdot 181^{2} + 173\cdot 181^{3} + 92\cdot 181^{4} + 10\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 162 + 169\cdot 181 + 65\cdot 181^{2} + 16\cdot 181^{3} + 19\cdot 181^{4} + 129\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,5)(6,8)$ |
| $(1,7,8,4,2,3,6,5)$ |
| $(3,7)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-4$ |
| $2$ | $2$ | $(3,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(4,5)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,8,2,6)(3,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,8,2,6)(3,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,7,8,4,2,3,6,5)$ | $0$ |
| $4$ | $8$ | $(1,4,6,7,2,5,8,3)$ | $0$ |
| $4$ | $8$ | $(1,7,8,5,2,3,6,4)$ | $0$ |
| $4$ | $8$ | $(1,5,6,7,2,4,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
