Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 10\cdot 61 + 44\cdot 61^{2} + 47\cdot 61^{3} + 58\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 52\cdot 61 + 34\cdot 61^{2} + 59\cdot 61^{3} + 57\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 35\cdot 61 + 8\cdot 61^{2} + 19\cdot 61^{3} + 2\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 16\cdot 61 + 60\cdot 61^{2} + 30\cdot 61^{3} + 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 23\cdot 61 + 15\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 11\cdot 61 + 17\cdot 61^{2} + 28\cdot 61^{3} + 25\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 56\cdot 61 + 43\cdot 61^{2} + 50\cdot 61^{3} + 29\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 38\cdot 61 + 34\cdot 61^{2} + 53\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,5,4,6)(2,8,3,7)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,5,4,6)(2,8,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
