Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 13\cdot 61 + 12\cdot 61^{2} + 45\cdot 61^{3} + 46\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 3\cdot 61 + 58\cdot 61^{2} + 7\cdot 61^{3} + 35\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 49\cdot 61 + 4\cdot 61^{2} + 52\cdot 61^{3} + 48\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 + 33\cdot 61 + 27\cdot 61^{2} + 14\cdot 61^{3} + 44\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 17\cdot 61 + 2\cdot 61^{2} + 3\cdot 61^{3} + 53\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 39\cdot 61 + 50\cdot 61^{2} + 14\cdot 61^{3} + 37\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 58 + 50\cdot 61 + 56\cdot 61^{2} + 58\cdot 61^{3} + 45\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 34\cdot 61 + 31\cdot 61^{2} + 47\cdot 61^{3} + 54\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,6)(4,7)(5,8)$ |
| $(1,4)(2,7)(3,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,8)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,7)(3,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,5)(2,4,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
