Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 367 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 66 + 121\cdot 367 + 254\cdot 367^{2} + 152\cdot 367^{3} + 344\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 199 + 73\cdot 367 + 237\cdot 367^{2} + 116\cdot 367^{3} + 30\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 214 + 291\cdot 367 + 276\cdot 367^{2} + 132\cdot 367^{3} + 43\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 286 + 116\cdot 367 + 75\cdot 367^{2} + 161\cdot 367^{3} + 69\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 336 + 130\cdot 367 + 257\cdot 367^{2} + 170\cdot 367^{3} + 246\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
