Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 106\cdot 151 + 74\cdot 151^{2} + 24\cdot 151^{3} + 57\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 34\cdot 151 + 75\cdot 151^{2} + 139\cdot 151^{3} + 29\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 46\cdot 151 + 4\cdot 151^{2} + 66\cdot 151^{3} + 86\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 88\cdot 151 + 33\cdot 151^{2} + 75\cdot 151^{3} + 74\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 95 + 62\cdot 151 + 117\cdot 151^{2} + 75\cdot 151^{3} + 76\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 114 + 104\cdot 151 + 146\cdot 151^{2} + 84\cdot 151^{3} + 64\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 115 + 116\cdot 151 + 75\cdot 151^{2} + 11\cdot 151^{3} + 121\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 127 + 44\cdot 151 + 76\cdot 151^{2} + 126\cdot 151^{3} + 93\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
