Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 20\cdot 149 + 54\cdot 149^{2} + 3\cdot 149^{3} + 130\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 83\cdot 149 + 76\cdot 149^{2} + 130\cdot 149^{3} + 58\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 113 + 2\cdot 149 + 127\cdot 149^{2} + 108\cdot 149^{3} + 36\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 116 + 42\cdot 149 + 40\cdot 149^{2} + 55\cdot 149^{3} + 72\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)$ |
| $(1,3)(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,2)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,2,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
