Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 45\cdot 61 + 2\cdot 61^{2} + 15\cdot 61^{3} + 47\cdot 61^{4} + 13\cdot 61^{5} + 54\cdot 61^{6} + 24\cdot 61^{7} + 37\cdot 61^{8} + 53\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 30\cdot 61 + 40\cdot 61^{2} + 48\cdot 61^{3} + 29\cdot 61^{4} + 34\cdot 61^{5} + 59\cdot 61^{6} + 56\cdot 61^{7} + 58\cdot 61^{8} + 2\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 21\cdot 61 + 32\cdot 61^{2} + 40\cdot 61^{3} + 19\cdot 61^{4} + 21\cdot 61^{5} + 41\cdot 61^{6} + 29\cdot 61^{7} + 34\cdot 61^{8} + 54\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 38\cdot 61 + 2\cdot 61^{2} + 18\cdot 61^{3} + 39\cdot 61^{4} + 2\cdot 61^{5} + 40\cdot 61^{6} + 20\cdot 61^{7} + 9\cdot 61^{8} + 46\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 22\cdot 61 + 58\cdot 61^{2} + 42\cdot 61^{3} + 21\cdot 61^{4} + 58\cdot 61^{5} + 20\cdot 61^{6} + 40\cdot 61^{7} + 51\cdot 61^{8} + 14\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 39\cdot 61 + 28\cdot 61^{2} + 20\cdot 61^{3} + 41\cdot 61^{4} + 39\cdot 61^{5} + 19\cdot 61^{6} + 31\cdot 61^{7} + 26\cdot 61^{8} + 6\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 30\cdot 61 + 20\cdot 61^{2} + 12\cdot 61^{3} + 31\cdot 61^{4} + 26\cdot 61^{5} + 61^{6} + 4\cdot 61^{7} + 2\cdot 61^{8} + 58\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 48 + 15\cdot 61 + 58\cdot 61^{2} + 45\cdot 61^{3} + 13\cdot 61^{4} + 47\cdot 61^{5} + 6\cdot 61^{6} + 36\cdot 61^{7} + 23\cdot 61^{8} + 7\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,7)(2,8)(4,5)$ |
| $(1,7,8,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
