Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 28\cdot 61 + 6\cdot 61^{2} + 18\cdot 61^{3} + 22\cdot 61^{4} + 17\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 37\cdot 61 + 25\cdot 61^{2} + 53\cdot 61^{3} + 11\cdot 61^{4} + 27\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 31\cdot 61 + 27\cdot 61^{2} + 30\cdot 61^{3} + 22\cdot 61^{4} + 49\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 5\cdot 61 + 7\cdot 61^{2} + 2\cdot 61^{3} + 14\cdot 61^{4} + 9\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 55\cdot 61 + 53\cdot 61^{2} + 58\cdot 61^{3} + 46\cdot 61^{4} + 51\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 29\cdot 61 + 33\cdot 61^{2} + 30\cdot 61^{3} + 38\cdot 61^{4} + 11\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 + 23\cdot 61 + 35\cdot 61^{2} + 7\cdot 61^{3} + 49\cdot 61^{4} + 33\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 54 + 32\cdot 61 + 54\cdot 61^{2} + 42\cdot 61^{3} + 38\cdot 61^{4} + 43\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
