Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 24\cdot 61 + 9\cdot 61^{2} + 60\cdot 61^{3} + 11\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 54\cdot 61 + 47\cdot 61^{2} + 19\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 45\cdot 61 + 5\cdot 61^{2} + 26\cdot 61^{3} + 8\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 36\cdot 61^{2} + 7\cdot 61^{3} + 14\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 60\cdot 61 + 20\cdot 61^{2} + 3\cdot 61^{3} + 42\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 47\cdot 61 + 35\cdot 61^{2} + 23\cdot 61^{3} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 52 + 24\cdot 61 + 50\cdot 61^{2} + 53\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 + 47\cdot 61 + 37\cdot 61^{2} + 41\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,7)(5,6)$ |
| $(1,4)(2,5)(3,7)(6,8)$ |
| $(1,5,3,6)(2,7,8,4)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,2,3,8)(4,5,7,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,3,2)(4,6,7,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,3,6)(2,7,8,4)$ | $0$ |
| $2$ | $4$ | $(1,8,3,2)(4,5,7,6)$ | $0$ |
| $2$ | $4$ | $(1,4,3,7)(2,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
