Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 112\cdot 149 + 21\cdot 149^{2} + 87\cdot 149^{3} + 51\cdot 149^{4} + 95\cdot 149^{5} + 95\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 127\cdot 149 + 37\cdot 149^{2} + 134\cdot 149^{3} + 4\cdot 149^{4} + 84\cdot 149^{5} + 32\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 17\cdot 149 + 17\cdot 149^{2} + 135\cdot 149^{3} + 148\cdot 149^{4} + 10\cdot 149^{5} + 100\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 85\cdot 149 + 57\cdot 149^{2} + 9\cdot 149^{3} + 60\cdot 149^{4} + 116\cdot 149^{5} + 112\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 103 + 63\cdot 149 + 91\cdot 149^{2} + 139\cdot 149^{3} + 88\cdot 149^{4} + 32\cdot 149^{5} + 36\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 134 + 131\cdot 149 + 131\cdot 149^{2} + 13\cdot 149^{3} + 138\cdot 149^{5} + 48\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 137 + 21\cdot 149 + 111\cdot 149^{2} + 14\cdot 149^{3} + 144\cdot 149^{4} + 64\cdot 149^{5} + 116\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 142 + 36\cdot 149 + 127\cdot 149^{2} + 61\cdot 149^{3} + 97\cdot 149^{4} + 53\cdot 149^{5} + 53\cdot 149^{6} +O\left(149^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(2,7)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,5,3,7)(2,8,4,6)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,3,7)(2,8,4,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,3,5)(2,6,4,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,6,2)(3,7,8,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,6,4)(3,5,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
