Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 30\cdot 61 + 11\cdot 61^{2} + 10\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 8\cdot 61^{2} + 57\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 3\cdot 61 + 34\cdot 61^{2} + 52\cdot 61^{3} + 10\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 4\cdot 61^{2} + 23\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 3\cdot 61 + 25\cdot 61^{2} + 38\cdot 61^{3} + 5\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 59\cdot 61 + 24\cdot 61^{2} + 14\cdot 61^{3} + 52\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 24\cdot 61 + 17\cdot 61^{2} + 13\cdot 61^{3} + 57\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 43 + 58\cdot 61^{2} + 34\cdot 61^{3} + 59\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,5,7,4,3,6)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
| $(1,8,7,3)(2,5,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,7)(2,4)(3,8)(5,6)$ |
$-1$ |
$-1$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,8,7,3)(2,5,4,6)$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
| $1$ |
$4$ |
$(1,3,7,8)(2,6,4,5)$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
| $1$ |
$8$ |
$(1,2,8,5,7,4,3,6)$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,5,3,2,7,6,8,4)$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
| $1$ |
$8$ |
$(1,4,8,6,7,2,3,5)$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,6,3,4,7,5,8,2)$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
