Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 46\cdot 61 + 45\cdot 61^{2} + 22\cdot 61^{3} + 43\cdot 61^{4} + 47\cdot 61^{5} + 12\cdot 61^{6} + 12\cdot 61^{7} + 60\cdot 61^{8} + 2\cdot 61^{9} +O\left(61^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 9\cdot 61 + 7\cdot 61^{2} + 47\cdot 61^{3} + 21\cdot 61^{4} + 24\cdot 61^{5} + 58\cdot 61^{6} + 60\cdot 61^{7} + 12\cdot 61^{8} + 16\cdot 61^{9} + 43\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 41\cdot 61 + 36\cdot 61^{2} + 43\cdot 61^{3} + 46\cdot 61^{4} + 7\cdot 61^{5} + 17\cdot 61^{6} + 57\cdot 61^{7} + 39\cdot 61^{8} + 42\cdot 61^{9} + 28\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 43\cdot 61 + 37\cdot 61^{2} + 10\cdot 61^{3} + 40\cdot 61^{4} + 4\cdot 61^{5} + 22\cdot 61^{6} + 42\cdot 61^{7} + 21\cdot 61^{8} + 20\cdot 61^{9} + 38\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 17\cdot 61 + 23\cdot 61^{2} + 50\cdot 61^{3} + 20\cdot 61^{4} + 56\cdot 61^{5} + 38\cdot 61^{6} + 18\cdot 61^{7} + 39\cdot 61^{8} + 40\cdot 61^{9} + 22\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 19\cdot 61 + 24\cdot 61^{2} + 17\cdot 61^{3} + 14\cdot 61^{4} + 53\cdot 61^{5} + 43\cdot 61^{6} + 3\cdot 61^{7} + 21\cdot 61^{8} + 18\cdot 61^{9} + 32\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 51\cdot 61 + 53\cdot 61^{2} + 13\cdot 61^{3} + 39\cdot 61^{4} + 36\cdot 61^{5} + 2\cdot 61^{6} + 48\cdot 61^{8} + 44\cdot 61^{9} + 17\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 48 + 14\cdot 61 + 15\cdot 61^{2} + 38\cdot 61^{3} + 17\cdot 61^{4} + 13\cdot 61^{5} + 48\cdot 61^{6} + 48\cdot 61^{7} + 58\cdot 61^{9} + 60\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,3,2,8,4,6,7)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2,8,7)(3,5,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,3)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $4$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
