Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 93\cdot 149 + 147\cdot 149^{2} + 14\cdot 149^{3} + 126\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 112\cdot 149 + 28\cdot 149^{2} + 56\cdot 149^{3} + 21\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 5\cdot 149 + 59\cdot 149^{2} + 102\cdot 149^{3} + 126\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 + 87\cdot 149 + 62\cdot 149^{2} + 124\cdot 149^{3} + 23\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 85 + 61\cdot 149 + 86\cdot 149^{2} + 24\cdot 149^{3} + 125\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 102 + 143\cdot 149 + 89\cdot 149^{2} + 46\cdot 149^{3} + 22\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 128 + 36\cdot 149 + 120\cdot 149^{2} + 92\cdot 149^{3} + 127\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 132 + 55\cdot 149 + 149^{2} + 134\cdot 149^{3} + 22\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,3)(5,6,8,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,3)(5,6,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
