Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 + 74\cdot 149 + 88\cdot 149^{2} + 88\cdot 149^{3} + 11\cdot 149^{4} + 31\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 + 123\cdot 149 + 25\cdot 149^{2} + 119\cdot 149^{3} + 9\cdot 149^{4} + 124\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 73 + 7\cdot 149 + 48\cdot 149^{2} + 139\cdot 149^{3} + 126\cdot 149^{4} + 46\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 87 + 104\cdot 149 + 101\cdot 149^{2} + 7\cdot 149^{3} + 14\cdot 149^{4} + 81\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 91 + 103\cdot 149 + 10\cdot 149^{2} + 95\cdot 149^{3} + 72\cdot 149^{4} + 90\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 129 + 139\cdot 149 + 124\cdot 149^{2} + 73\cdot 149^{3} + 137\cdot 149^{4} + 121\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 130 + 71\cdot 149 + 57\cdot 149^{2} + 136\cdot 149^{3} + 99\cdot 149^{4} + 132\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 142 + 119\cdot 149 + 138\cdot 149^{2} + 84\cdot 149^{3} + 123\cdot 149^{4} + 116\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,3)(5,7)(6,8)$ |
| $(5,8)(6,7)$ |
| $(1,7)(2,6)(3,5)(4,8)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
| $(1,6,4,8)(2,7,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $-4$ |
| $2$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,6)(3,5)(4,8)$ | $0$ |
| $4$ | $4$ | $(1,6,4,8)(2,7,3,5)$ | $0$ |
| $4$ | $4$ | $(1,8,4,6)(2,5,3,7)$ | $0$ |
| $4$ | $4$ | $(1,8,2,5)(3,6,4,7)$ | $0$ |
| $4$ | $4$ | $(1,2)(5,7,8,6)$ | $0$ |
| $4$ | $4$ | $(1,2)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
