Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 143\cdot 151 + 43\cdot 151^{2} + 43\cdot 151^{3} + 39\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 102\cdot 151 + 21\cdot 151^{2} + 144\cdot 151^{3} + 114\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 77 + 106\cdot 151 + 25\cdot 151^{2} + 29\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 88 + 94\cdot 151 + 119\cdot 151^{2} + 62\cdot 151^{3} + 95\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 93 + 58\cdot 151 + 2\cdot 151^{2} + 81\cdot 151^{3} + 94\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 99 + 17\cdot 151 + 131\cdot 151^{2} + 30\cdot 151^{3} + 19\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 109 + 46\cdot 151 + 7\cdot 151^{2} + 14\cdot 151^{3} + 148\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 120 + 34\cdot 151 + 101\cdot 151^{2} + 47\cdot 151^{3} + 68\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,3,6,8)(2,4,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,7)(4,8)(5,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,6,8)(2,4,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
