Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 15.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 15\cdot 61 + 60\cdot 61^{2} + 2\cdot 61^{3} + 43\cdot 61^{4} + 33\cdot 61^{5} + 20\cdot 61^{6} + 18\cdot 61^{7} + 52\cdot 61^{8} + 8\cdot 61^{9} + 15\cdot 61^{10} + 2\cdot 61^{11} + 25\cdot 61^{12} + 22\cdot 61^{13} + 20\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 22\cdot 61 + 20\cdot 61^{2} + 17\cdot 61^{3} + 29\cdot 61^{4} + 32\cdot 61^{5} + 12\cdot 61^{6} + 43\cdot 61^{7} + 39\cdot 61^{8} + 37\cdot 61^{9} + 53\cdot 61^{10} + 7\cdot 61^{11} + 39\cdot 61^{12} + 12\cdot 61^{13} + 54\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 18\cdot 61 + 16\cdot 61^{2} + 26\cdot 61^{3} + 29\cdot 61^{4} + 9\cdot 61^{5} + 5\cdot 61^{6} + 5\cdot 61^{7} + 7\cdot 61^{8} + 15\cdot 61^{9} + 55\cdot 61^{10} + 2\cdot 61^{11} + 6\cdot 61^{12} + 2\cdot 61^{13} + 32\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 45\cdot 61 + 6\cdot 61^{2} + 41\cdot 61^{3} + 7\cdot 61^{4} + 11\cdot 61^{5} + 12\cdot 61^{6} + 3\cdot 61^{7} + 37\cdot 61^{8} + 22\cdot 61^{9} + 10\cdot 61^{10} + 38\cdot 61^{11} + 5\cdot 61^{12} + 43\cdot 61^{13} + 35\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 15\cdot 61 + 54\cdot 61^{2} + 19\cdot 61^{3} + 53\cdot 61^{4} + 49\cdot 61^{5} + 48\cdot 61^{6} + 57\cdot 61^{7} + 23\cdot 61^{8} + 38\cdot 61^{9} + 50\cdot 61^{10} + 22\cdot 61^{11} + 55\cdot 61^{12} + 17\cdot 61^{13} + 25\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 42\cdot 61 + 44\cdot 61^{2} + 34\cdot 61^{3} + 31\cdot 61^{4} + 51\cdot 61^{5} + 55\cdot 61^{6} + 55\cdot 61^{7} + 53\cdot 61^{8} + 45\cdot 61^{9} + 5\cdot 61^{10} + 58\cdot 61^{11} + 54\cdot 61^{12} + 58\cdot 61^{13} + 28\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 38\cdot 61 + 40\cdot 61^{2} + 43\cdot 61^{3} + 31\cdot 61^{4} + 28\cdot 61^{5} + 48\cdot 61^{6} + 17\cdot 61^{7} + 21\cdot 61^{8} + 23\cdot 61^{9} + 7\cdot 61^{10} + 53\cdot 61^{11} + 21\cdot 61^{12} + 48\cdot 61^{13} + 6\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 45\cdot 61 + 58\cdot 61^{3} + 17\cdot 61^{4} + 27\cdot 61^{5} + 40\cdot 61^{6} + 42\cdot 61^{7} + 8\cdot 61^{8} + 52\cdot 61^{9} + 45\cdot 61^{10} + 58\cdot 61^{11} + 35\cdot 61^{12} + 38\cdot 61^{13} + 40\cdot 61^{14} +O\left(61^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(2,3)(4,5)(6,7)$ |
| $(1,2,4,3,8,7,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,2,4,3,8,7,5,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
