Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 16\cdot 59 + 4\cdot 59^{2} + 5\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 14\cdot 59 + 17\cdot 59^{2} + 49\cdot 59^{3} + 14\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 10\cdot 59 + 17\cdot 59^{2} + 11\cdot 59^{3} + 32\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 26\cdot 59 + 37\cdot 59^{2} + 39\cdot 59^{3} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 50\cdot 59 + 41\cdot 59^{2} + 17\cdot 59^{3} + 6\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,3,5,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,3,5,2,4)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,5,4,3,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,4,2,5,3)$ | $\zeta_{5}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
