Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 24\cdot 59 + 25\cdot 59^{2} + 3\cdot 59^{3} + 46\cdot 59^{4} + 18\cdot 59^{5} + 14\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 17\cdot 59 + 19\cdot 59^{2} + 6\cdot 59^{3} + 46\cdot 59^{4} + 10\cdot 59^{5} + 54\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 45\cdot 59 + 21\cdot 59^{2} + 59^{3} + 41\cdot 59^{4} + 4\cdot 59^{5} + 40\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 31\cdot 59 + 55\cdot 59^{2} + 56\cdot 59^{3} + 5\cdot 59^{4} + 44\cdot 59^{5} + 23\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 + 27\cdot 59 + 3\cdot 59^{2} + 2\cdot 59^{3} + 53\cdot 59^{4} + 14\cdot 59^{5} + 35\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 13\cdot 59 + 37\cdot 59^{2} + 57\cdot 59^{3} + 17\cdot 59^{4} + 54\cdot 59^{5} + 18\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 41\cdot 59 + 39\cdot 59^{2} + 52\cdot 59^{3} + 12\cdot 59^{4} + 48\cdot 59^{5} + 4\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 34\cdot 59 + 33\cdot 59^{2} + 55\cdot 59^{3} + 12\cdot 59^{4} + 40\cdot 59^{5} + 44\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,5,7,3,8,4,2,6)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,7,3,8,4,2,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,2,5,8,6,7,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,7,4,8,6,2,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,2,3,8,5,7,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
