Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 14\cdot 229 + 44\cdot 229^{2} + 74\cdot 229^{3} + 90\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 136 + 119\cdot 229 + 111\cdot 229^{2} + 25\cdot 229^{3} + 169\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 148 + 171\cdot 229 + 43\cdot 229^{2} + 72\cdot 229^{3} + 42\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 162 + 17\cdot 229 + 167\cdot 229^{2} + 27\cdot 229^{3} + 168\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 226 + 134\cdot 229 + 91\cdot 229^{2} + 29\cdot 229^{3} + 217\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
