Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 359 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 79 + 247\cdot 359 + 267\cdot 359^{2} + 339\cdot 359^{3} + 342\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 81 + 335\cdot 359 + 95\cdot 359^{2} + 98\cdot 359^{3} + 281\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 125 + 128\cdot 359 + 9\cdot 359^{2} + 319\cdot 359^{3} + 352\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 145 + 121\cdot 359 + 226\cdot 359^{2} + 175\cdot 359^{3} + 278\cdot 359^{4} +O\left(359^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 290 + 244\cdot 359 + 118\cdot 359^{2} + 144\cdot 359^{3} + 180\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
