Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 4\cdot 61 + 41\cdot 61^{2} + 9\cdot 61^{3} + 14\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 3\cdot 61 + 57\cdot 61^{2} + 16\cdot 61^{3} + 17\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 29\cdot 61 + 36\cdot 61^{2} + 38\cdot 61^{3} + 17\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 19\cdot 61 + 58\cdot 61^{2} + 49\cdot 61^{3} + 59\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 34\cdot 61 + 26\cdot 61^{2} + 45\cdot 61^{3} + 30\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 10\cdot 61 + 6\cdot 61^{2} + 21\cdot 61^{3} + 41\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 34\cdot 61 + 20\cdot 61^{2} + 40\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 47\cdot 61 + 58\cdot 61^{2} + 21\cdot 61^{3} + 6\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(2,4)(3,8)$ |
| $(1,5)(2,4)(3,8)(6,7)$ |
| $(1,3)(2,7)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(2,4)(3,8)$ | $0$ |
| $1$ | $4$ | $(1,6,5,7)(2,3,4,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,5,6)(2,8,4,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,5,3)(2,7,4,6)$ | $0$ |
| $2$ | $4$ | $(1,6,5,7)(2,8,4,3)$ | $0$ |
| $2$ | $4$ | $(1,4,5,2)(3,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
