Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 14.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 58\cdot 61 + 48\cdot 61^{2} + 5\cdot 61^{3} + 30\cdot 61^{4} + 35\cdot 61^{5} + 44\cdot 61^{6} + 27\cdot 61^{7} + 32\cdot 61^{8} + 28\cdot 61^{9} + 35\cdot 61^{10} + 16\cdot 61^{11} + 31\cdot 61^{12} + 27\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 58\cdot 61 + 22\cdot 61^{2} + 17\cdot 61^{3} + 52\cdot 61^{4} + 51\cdot 61^{5} + 34\cdot 61^{6} + 42\cdot 61^{7} + 32\cdot 61^{8} + 18\cdot 61^{9} + 47\cdot 61^{10} + 7\cdot 61^{11} + 50\cdot 61^{12} + 26\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 5\cdot 61 + 60\cdot 61^{2} + 28\cdot 61^{3} + 37\cdot 61^{4} + 12\cdot 61^{5} + 48\cdot 61^{6} + 21\cdot 61^{7} + 23\cdot 61^{8} + 31\cdot 61^{9} + 31\cdot 61^{10} + 21\cdot 61^{11} + 11\cdot 61^{12} + 54\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 2\cdot 61 + 39\cdot 61^{2} + 33\cdot 61^{3} + 61^{5} + 24\cdot 61^{6} + 24\cdot 61^{7} + 41\cdot 61^{8} + 13\cdot 61^{9} + 36\cdot 61^{10} + 8\cdot 61^{11} + 31\cdot 61^{12} + 46\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 58\cdot 61 + 21\cdot 61^{2} + 27\cdot 61^{3} + 60\cdot 61^{4} + 59\cdot 61^{5} + 36\cdot 61^{6} + 36\cdot 61^{7} + 19\cdot 61^{8} + 47\cdot 61^{9} + 24\cdot 61^{10} + 52\cdot 61^{11} + 29\cdot 61^{12} + 14\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 + 55\cdot 61 + 32\cdot 61^{3} + 23\cdot 61^{4} + 48\cdot 61^{5} + 12\cdot 61^{6} + 39\cdot 61^{7} + 37\cdot 61^{8} + 29\cdot 61^{9} + 29\cdot 61^{10} + 39\cdot 61^{11} + 49\cdot 61^{12} + 6\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 48 + 2\cdot 61 + 38\cdot 61^{2} + 43\cdot 61^{3} + 8\cdot 61^{4} + 9\cdot 61^{5} + 26\cdot 61^{6} + 18\cdot 61^{7} + 28\cdot 61^{8} + 42\cdot 61^{9} + 13\cdot 61^{10} + 53\cdot 61^{11} + 10\cdot 61^{12} + 34\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 2\cdot 61 + 12\cdot 61^{2} + 55\cdot 61^{3} + 30\cdot 61^{4} + 25\cdot 61^{5} + 16\cdot 61^{6} + 33\cdot 61^{7} + 28\cdot 61^{8} + 32\cdot 61^{9} + 25\cdot 61^{10} + 44\cdot 61^{11} + 29\cdot 61^{12} + 33\cdot 61^{13} +O\left(61^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,7,5,8,3,2,4)$ |
| $(3,6)(4,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $4$ | $8$ | $(1,6,7,5,8,3,2,4)$ | $0$ |
| $4$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $0$ |
| $4$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
| $4$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
