Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 16\cdot 61 + 22\cdot 61^{2} + 51\cdot 61^{3} + 10\cdot 61^{4} + 56\cdot 61^{5} + 40\cdot 61^{7} + 46\cdot 61^{8} + 11\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 36\cdot 61 + 30\cdot 61^{2} + 30\cdot 61^{3} + 49\cdot 61^{4} + 9\cdot 61^{5} + 37\cdot 61^{6} + 23\cdot 61^{7} + 59\cdot 61^{8} + 9\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 25\cdot 61 + 5\cdot 61^{2} + 2\cdot 61^{3} + 55\cdot 61^{4} + 46\cdot 61^{5} + 28\cdot 61^{6} + 57\cdot 61^{7} + 60\cdot 61^{8} + 60\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 44\cdot 61 + 11\cdot 61^{2} + 43\cdot 61^{3} + 13\cdot 61^{4} + 38\cdot 61^{5} + 13\cdot 61^{6} + 29\cdot 61^{7} + 35\cdot 61^{8} + 30\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 + 16\cdot 61 + 49\cdot 61^{2} + 17\cdot 61^{3} + 47\cdot 61^{4} + 22\cdot 61^{5} + 47\cdot 61^{6} + 31\cdot 61^{7} + 25\cdot 61^{8} + 30\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 43 + 35\cdot 61 + 55\cdot 61^{2} + 58\cdot 61^{3} + 5\cdot 61^{4} + 14\cdot 61^{5} + 32\cdot 61^{6} + 3\cdot 61^{7} +O\left(61^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 24\cdot 61 + 30\cdot 61^{2} + 30\cdot 61^{3} + 11\cdot 61^{4} + 51\cdot 61^{5} + 23\cdot 61^{6} + 37\cdot 61^{7} + 61^{8} + 51\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 + 44\cdot 61 + 38\cdot 61^{2} + 9\cdot 61^{3} + 50\cdot 61^{4} + 4\cdot 61^{5} + 60\cdot 61^{6} + 20\cdot 61^{7} + 14\cdot 61^{8} + 49\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,5,6,8,2,4,3)$ |
| $(1,5,8,4)(2,6,7,3)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,7,5,6,8,2,4,3)$ | $0$ |
| $2$ | $8$ | $(1,6,4,7,8,3,5,2)$ | $0$ |
| $2$ | $8$ | $(1,6,5,2,8,3,4,7)$ | $0$ |
| $2$ | $8$ | $(1,2,4,6,8,7,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
