Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 43\cdot 61 + 6\cdot 61^{2} + 19\cdot 61^{3} + 58\cdot 61^{4} + 56\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 47\cdot 61 + 9\cdot 61^{2} + 61^{3} + 33\cdot 61^{4} + 21\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 51\cdot 61 + 20\cdot 61^{2} + 56\cdot 61^{3} + 31\cdot 61^{4} + 44\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 55\cdot 61 + 23\cdot 61^{2} + 38\cdot 61^{3} + 6\cdot 61^{4} + 9\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 5\cdot 61 + 37\cdot 61^{2} + 22\cdot 61^{3} + 54\cdot 61^{4} + 51\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 9\cdot 61 + 40\cdot 61^{2} + 4\cdot 61^{3} + 29\cdot 61^{4} + 16\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 13\cdot 61 + 51\cdot 61^{2} + 59\cdot 61^{3} + 27\cdot 61^{4} + 39\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 17\cdot 61 + 54\cdot 61^{2} + 41\cdot 61^{3} + 2\cdot 61^{4} + 4\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,4)(2,3,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
