Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 24\cdot 59 + 5\cdot 59^{2} + 32\cdot 59^{3} + 29\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 48\cdot 59 + 45\cdot 59^{2} + 14\cdot 59^{3} + 14\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 39\cdot 59 + 15\cdot 59^{2} + 58\cdot 59^{3} + 29\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 6\cdot 59 + 51\cdot 59^{2} + 12\cdot 59^{3} + 44\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 42\cdot 59 + 3\cdot 59^{2} + 45\cdot 59^{3} + 15\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 2\cdot 59 + 8\cdot 59^{2} + 35\cdot 59^{3} + 12\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 + 9\cdot 59 + 18\cdot 59^{2} + 42\cdot 59^{3} + 28\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 3\cdot 59 + 29\cdot 59^{2} + 54\cdot 59^{3} + 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,3,6,4,5,2,7)$ |
| $(1,3,4,2)(5,7,8,6)$ |
| $(5,8)(6,7)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $1$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,4,2)(5,6,8,7)$ | $0$ |
| $2$ | $8$ | $(1,8,3,6,4,5,2,7)$ | $0$ |
| $2$ | $8$ | $(1,6,2,8,4,7,3,5)$ | $0$ |
| $2$ | $8$ | $(1,5,2,6,4,8,3,7)$ | $0$ |
| $2$ | $8$ | $(1,6,3,5,4,7,2,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
