Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 29\cdot 61 + 33\cdot 61^{2} + 51\cdot 61^{3} + 17\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 36\cdot 61 + 49\cdot 61^{2} + 35\cdot 61^{3} + 29\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 3\cdot 61 + 45\cdot 61^{2} + 54\cdot 61^{3} + 5\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 46\cdot 61 + 29\cdot 61^{2} + 39\cdot 61^{3} + 44\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 7\cdot 61 + 42\cdot 61^{3} + 15\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 3\cdot 61 + 47\cdot 61^{2} + 46\cdot 61^{3} + 55\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 + 20\cdot 61 + 41\cdot 61^{2} + 42\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 51 + 35\cdot 61 + 58\cdot 61^{2} + 52\cdot 61^{3} + 41\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
