Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 18\cdot 149 + 138\cdot 149^{2} + 32\cdot 149^{3} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 52\cdot 149 + 27\cdot 149^{2} + 126\cdot 149^{3} + 57\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 68\cdot 149 + 136\cdot 149^{2} + 122\cdot 149^{3} + 7\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 + 10\cdot 149 + 145\cdot 149^{2} + 15\cdot 149^{3} + 83\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 62 + 94\cdot 149 + 99\cdot 149^{2} + 140\cdot 149^{3} + 20\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 101 + 77\cdot 149 + 51\cdot 149^{2} + 110\cdot 149^{3} + 31\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 140 + 107\cdot 149 + 45\cdot 149^{2} + 111\cdot 149^{3} + 11\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 144 + 17\cdot 149 + 101\cdot 149^{2} + 84\cdot 149^{3} + 84\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,6,7,8)$ |
| $(1,5)(2,7)(3,8)(4,6)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,8)(4,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,6)(3,7)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,4)(5,6,7,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
