Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 28\cdot 61 + 2\cdot 61^{2} + 30\cdot 61^{3} + 34\cdot 61^{4} + 10\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 46\cdot 61 + 60\cdot 61^{2} + 53\cdot 61^{3} + 29\cdot 61^{4} + 51\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 50\cdot 61 + 13\cdot 61^{2} + 13\cdot 61^{3} + 39\cdot 61^{4} + 25\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 7\cdot 61 + 11\cdot 61^{2} + 37\cdot 61^{3} + 34\cdot 61^{4} + 5\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 + 53\cdot 61 + 49\cdot 61^{2} + 23\cdot 61^{3} + 26\cdot 61^{4} + 55\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 10\cdot 61 + 47\cdot 61^{2} + 47\cdot 61^{3} + 21\cdot 61^{4} + 35\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 48 + 14\cdot 61 + 7\cdot 61^{3} + 31\cdot 61^{4} + 9\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 49 + 32\cdot 61 + 58\cdot 61^{2} + 30\cdot 61^{3} + 26\cdot 61^{4} + 50\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,4,6)(2,5,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
