Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 25\cdot 61 + 32\cdot 61^{2} + 55\cdot 61^{3} + 26\cdot 61^{4} + 17\cdot 61^{5} + 39\cdot 61^{6} + 50\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 35\cdot 61 + 2\cdot 61^{2} + 26\cdot 61^{3} + 22\cdot 61^{4} + 13\cdot 61^{5} + 2\cdot 61^{6} + 18\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 7\cdot 61 + 36\cdot 61^{2} + 35\cdot 61^{3} + 23\cdot 61^{4} + 44\cdot 61^{5} + 25\cdot 61^{6} + 31\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 3\cdot 61 + 39\cdot 61^{3} + 3\cdot 61^{4} + 41\cdot 61^{5} + 53\cdot 61^{6} + 5\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 57\cdot 61 + 60\cdot 61^{2} + 21\cdot 61^{3} + 57\cdot 61^{4} + 19\cdot 61^{5} + 7\cdot 61^{6} + 55\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 + 53\cdot 61 + 24\cdot 61^{2} + 25\cdot 61^{3} + 37\cdot 61^{4} + 16\cdot 61^{5} + 35\cdot 61^{6} + 29\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 25\cdot 61 + 58\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} + 47\cdot 61^{5} + 58\cdot 61^{6} + 42\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 60 + 35\cdot 61 + 28\cdot 61^{2} + 5\cdot 61^{3} + 34\cdot 61^{4} + 43\cdot 61^{5} + 21\cdot 61^{6} + 10\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,6,7,4,8,3,2,5)$ |
| $(3,6)(4,5)$ |
| $(1,8)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(2,7)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,7,4,8,3,2,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,2,6,8,5,7,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,2,4,8,6,7,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,7,3,8,5,2,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
