Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 2\cdot 61 + 51\cdot 61^{2} + 8\cdot 61^{3} + 25\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 22\cdot 61 + 46\cdot 61^{2} + 16\cdot 61^{3} + 43\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 44\cdot 61 + 47\cdot 61^{2} + 46\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 25\cdot 61 + 49\cdot 61^{2} + 11\cdot 61^{3} + 3\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 4\cdot 61 + 31\cdot 61^{2} + 46\cdot 61^{3} + 9\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 51\cdot 61 + 40\cdot 61^{2} + 35\cdot 61^{3} + 12\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 31\cdot 61 + 17\cdot 61^{2} + 56\cdot 61^{3} + 45\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 60\cdot 61 + 20\cdot 61^{2} + 21\cdot 61^{3} + 3\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,7,8,6)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,8)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,8)(3,7)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,5)(4,7,8,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
