Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 39\cdot 61 + 24\cdot 61^{2} + 56\cdot 61^{3} + 33\cdot 61^{4} + 24\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 34\cdot 61 + 40\cdot 61^{2} + 39\cdot 61^{3} + 59\cdot 61^{4} + 4\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 30\cdot 61 + 10\cdot 61^{2} + 30\cdot 61^{3} + 28\cdot 61^{4} + 32\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 18\cdot 61 + 46\cdot 61^{2} + 56\cdot 61^{3} + 60\cdot 61^{4} + 59\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 42\cdot 61 + 14\cdot 61^{2} + 4\cdot 61^{3} + 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 30\cdot 61 + 50\cdot 61^{2} + 30\cdot 61^{3} + 32\cdot 61^{4} + 28\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 26\cdot 61 + 20\cdot 61^{2} + 21\cdot 61^{3} + 61^{4} + 56\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 21\cdot 61 + 36\cdot 61^{2} + 4\cdot 61^{3} + 27\cdot 61^{4} + 36\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,4)(5,8,7,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,8)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,4)(5,8,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
