Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 59\cdot 113 + 44\cdot 113^{2} + 88\cdot 113^{3} + 46\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 39\cdot 113 + 100\cdot 113^{2} + 38\cdot 113^{3} + 99\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 83\cdot 113 + 49\cdot 113^{2} + 112\cdot 113^{3} + 107\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 68\cdot 113 + 81\cdot 113^{2} + 13\cdot 113^{3} + 28\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 + 44\cdot 113 + 31\cdot 113^{2} + 99\cdot 113^{3} + 84\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 88 + 29\cdot 113 + 63\cdot 113^{2} + 5\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 101 + 73\cdot 113 + 12\cdot 113^{2} + 74\cdot 113^{3} + 13\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 104 + 53\cdot 113 + 68\cdot 113^{2} + 24\cdot 113^{3} + 66\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,2,5,3)(4,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,3)(4,6,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
