Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7\cdot 59 + 43\cdot 59^{2} + 36\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 7\cdot 59 + 53\cdot 59^{2} + 56\cdot 59^{3} + 57\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 15\cdot 59 + 17\cdot 59^{2} + 49\cdot 59^{3} + 48\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 52\cdot 59 + 17\cdot 59^{2} + 4\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 3\cdot 59 + 54\cdot 59^{2} + 8\cdot 59^{3} + 4\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 42\cdot 59 + 8\cdot 59^{2} + 27\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 56 + 43\cdot 59 + 41\cdot 59^{2} + 38\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 3\cdot 59 + 18\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,6,4,5,2,8,3)$ |
| $(1,8,5,6)(2,4,7,3)$ |
| $(1,5)(2,7)(3,4)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $-1$ |
| $1$ | $4$ | $(1,6,5,8)(2,3,7,4)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,8,5,6)(2,4,7,3)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,7,6,4,5,2,8,3)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,4,8,7,5,3,6,2)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,2,6,3,5,7,8,4)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,3,8,2,5,4,6,7)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
