Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 359 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 13\cdot 359 + 219\cdot 359^{2} + 48\cdot 359^{3} + 44\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 44\cdot 359 + 54\cdot 359^{2} + 88\cdot 359^{3} + 320\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 + 115\cdot 359 + 91\cdot 359^{2} + 142\cdot 359^{3} + 258\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 293 + 173\cdot 359 + 139\cdot 359^{2} + 62\cdot 359^{3} + 260\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 310 + 12\cdot 359 + 214\cdot 359^{2} + 17\cdot 359^{3} + 194\cdot 359^{4} +O\left(359^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
