Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 36\cdot 61 + 56\cdot 61^{2} + 40\cdot 61^{3} + 24\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 51\cdot 61 + 45\cdot 61^{2} + 45\cdot 61^{3} + 2\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 33\cdot 61 + 40\cdot 61^{2} + 38\cdot 61^{3} + 53\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 33\cdot 61 + 21\cdot 61^{2} + 27\cdot 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 27\cdot 61 + 39\cdot 61^{2} + 33\cdot 61^{3} + 10\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 27\cdot 61 + 20\cdot 61^{2} + 22\cdot 61^{3} + 7\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 9\cdot 61 + 15\cdot 61^{2} + 15\cdot 61^{3} + 58\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 + 24\cdot 61 + 4\cdot 61^{2} + 20\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
