Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 61 + 25\cdot 61^{2} + 16\cdot 61^{3} + 19\cdot 61^{4} + 55\cdot 61^{5} + 57\cdot 61^{6} + 61^{7} + 8\cdot 61^{8} + 38\cdot 61^{9} + 8\cdot 61^{10} + 5\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 23\cdot 61 + 3\cdot 61^{2} + 43\cdot 61^{3} + 2\cdot 61^{4} + 37\cdot 61^{5} + 23\cdot 61^{6} + 14\cdot 61^{7} + 39\cdot 61^{8} + 8\cdot 61^{9} + 16\cdot 61^{10} + 41\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 11\cdot 61 + 50\cdot 61^{2} + 11\cdot 61^{3} + 49\cdot 61^{4} + 27\cdot 61^{5} + 61^{6} + 29\cdot 61^{7} + 23\cdot 61^{8} + 48\cdot 61^{9} + 43\cdot 61^{10} + 30\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 40\cdot 61 + 61^{2} + 9\cdot 61^{3} + 48\cdot 61^{4} + 3\cdot 61^{5} + 35\cdot 61^{6} + 30\cdot 61^{7} + 41\cdot 61^{8} + 3\cdot 61^{9} + 12\cdot 61^{10} + 3\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 20\cdot 61 + 59\cdot 61^{2} + 51\cdot 61^{3} + 12\cdot 61^{4} + 57\cdot 61^{5} + 25\cdot 61^{6} + 30\cdot 61^{7} + 19\cdot 61^{8} + 57\cdot 61^{9} + 48\cdot 61^{10} + 57\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 49\cdot 61 + 10\cdot 61^{2} + 49\cdot 61^{3} + 11\cdot 61^{4} + 33\cdot 61^{5} + 59\cdot 61^{6} + 31\cdot 61^{7} + 37\cdot 61^{8} + 12\cdot 61^{9} + 17\cdot 61^{10} + 30\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 54 + 37\cdot 61 + 57\cdot 61^{2} + 17\cdot 61^{3} + 58\cdot 61^{4} + 23\cdot 61^{5} + 37\cdot 61^{6} + 46\cdot 61^{7} + 21\cdot 61^{8} + 52\cdot 61^{9} + 44\cdot 61^{10} + 19\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 + 59\cdot 61 + 35\cdot 61^{2} + 44\cdot 61^{3} + 41\cdot 61^{4} + 5\cdot 61^{5} + 3\cdot 61^{6} + 59\cdot 61^{7} + 52\cdot 61^{8} + 22\cdot 61^{9} + 52\cdot 61^{10} + 55\cdot 61^{11} +O\left(61^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
