Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 39\cdot 61 + 33\cdot 61^{2} + 22\cdot 61^{3} + 52\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 39\cdot 61 + 58\cdot 61^{2} + 46\cdot 61^{3} + 28\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 46\cdot 61 + 59\cdot 61^{2} + 16\cdot 61^{3} + 59\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 20\cdot 61 + 41\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 35\cdot 61 + 24\cdot 61^{2} + 8\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 59\cdot 61 + 42\cdot 61^{2} + 15\cdot 61^{3} + 20\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 2\cdot 61 + 45\cdot 61^{2} + 42\cdot 61^{3} + 14\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 40\cdot 61^{2} + 49\cdot 61^{3} + 38\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,6,3)(2,4,5,7)$ |
| $(1,7)(2,5)(4,6)$ |
| $(1,6)(2,5)(3,8)(4,7)$ |
| $(1,4,6,7)(2,3,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,5)(4,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,4,6,7)(2,3,5,8)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,8,6,3)(2,4,5,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,8,4,2,6,3,7,5)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,3,4,5,6,8,7,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
