Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 73\cdot 151 + 126\cdot 151^{2} + 149\cdot 151^{3} + 56\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 97\cdot 151 + 83\cdot 151^{2} + 127\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 111 + 132\cdot 151 + 10\cdot 151^{2} + 97\cdot 151^{3} + 135\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 145 + 149\cdot 151 + 80\cdot 151^{2} + 78\cdot 151^{3} + 84\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
