Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 24\cdot 59 + 53\cdot 59^{2} + 13\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 27\cdot 59 + 41\cdot 59^{2} + 22\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 46\cdot 59 + 47\cdot 59^{2} + 16\cdot 59^{3} + 57\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 3\cdot 59 + 35\cdot 59^{2} + 29\cdot 59^{3} + 13\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 20\cdot 59^{2} + 28\cdot 59^{3} + 20\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 5\cdot 59 + 20\cdot 59^{3} + 39\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 + 6\cdot 59 + 42\cdot 59^{2} + 8\cdot 59^{3} + 14\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 3\cdot 59 + 55\cdot 59^{2} + 36\cdot 59^{3} + 56\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,4,5,2,8,6,3)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,2,6)(3,7,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-1$ |
| $1$ | $4$ | $(1,4,2,6)(3,7,5,8)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,6,2,4)(3,8,5,7)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,7,4,5,2,8,6,3)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,5,6,7,2,3,4,8)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,8,4,3,2,7,6,5)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,3,6,8,2,5,4,7)$ | $-\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
