Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 56\cdot 61 + 25\cdot 61^{2} + 38\cdot 61^{3} + 7\cdot 61^{4} + 19\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 40\cdot 61 + 24\cdot 61^{2} + 36\cdot 61^{3} + 36\cdot 61^{4} + 36\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 58\cdot 61 + 46\cdot 61^{2} + 46\cdot 61^{3} + 46\cdot 61^{4} + 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 14\cdot 61 + 48\cdot 61^{2} + 4\cdot 61^{3} + 19\cdot 61^{4} + 58\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 46\cdot 61 + 12\cdot 61^{2} + 56\cdot 61^{3} + 41\cdot 61^{4} + 2\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 + 2\cdot 61 + 14\cdot 61^{2} + 14\cdot 61^{3} + 14\cdot 61^{4} + 59\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 + 20\cdot 61 + 36\cdot 61^{2} + 24\cdot 61^{3} + 24\cdot 61^{4} + 24\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 + 4\cdot 61 + 35\cdot 61^{2} + 22\cdot 61^{3} + 53\cdot 61^{4} + 41\cdot 61^{5} +O\left(61^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(3,6)(4,5)$ |
| $(1,7,8,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
