Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 33\cdot 113 + 39\cdot 113^{2} + 8\cdot 113^{3} + 78\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 110\cdot 113 + 77\cdot 113^{2} + 68\cdot 113^{3} + 62\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 + 88\cdot 113 + 23\cdot 113^{2} + 94\cdot 113^{3} + 70\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 52\cdot 113 + 62\cdot 113^{2} + 41\cdot 113^{3} + 55\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 31\cdot 113 + 7\cdot 113^{2} + 96\cdot 113^{3} + 64\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 108\cdot 113 + 45\cdot 113^{2} + 43\cdot 113^{3} + 49\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 83 + 31\cdot 113 + 78\cdot 113^{2} + 19\cdot 113^{3} + 43\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 94 + 108\cdot 113 + 3\cdot 113^{2} + 80\cdot 113^{3} + 27\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,3,5,7)(2,4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,5,7)(2,4,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
