Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 29\cdot 59 + 9\cdot 59^{2} + 4\cdot 59^{3} + 22\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 3\cdot 59 + 35\cdot 59^{2} + 51\cdot 59^{3} + 18\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 22\cdot 59 + 11\cdot 59^{2} + 41\cdot 59^{3} + 37\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 25\cdot 59 + 58\cdot 59^{2} + 21\cdot 59^{3} + 6\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 21\cdot 59 + 59^{2} + 44\cdot 59^{3} + 19\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 41\cdot 59 + 57\cdot 59^{2} + 39\cdot 59^{3} + 36\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 + 26\cdot 59 + 45\cdot 59^{2} + 47\cdot 59^{3} + 34\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 49 + 6\cdot 59 + 17\cdot 59^{2} + 44\cdot 59^{3} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,3)(4,8,6,7)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,6,5,7,2,4,3,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-1$ |
| $1$ | $4$ | $(1,5,2,3)(4,8,6,7)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,3,2,5)(4,7,6,8)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,6,5,7,2,4,3,8)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,7,3,6,2,8,5,4)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,5,8,2,6,3,7)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,8,3,4,2,7,5,6)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
