Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 98\cdot 113 + 12\cdot 113^{2} + 30\cdot 113^{3} + 98\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 + 44\cdot 113 + 7\cdot 113^{2} + 90\cdot 113^{3} + 46\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 112\cdot 113 + 79\cdot 113^{2} + 10\cdot 113^{3} + 60\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 54\cdot 113 + 38\cdot 113^{2} + 42\cdot 113^{3} + 104\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 + 58\cdot 113 + 74\cdot 113^{2} + 70\cdot 113^{3} + 8\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 76 + 33\cdot 113^{2} + 102\cdot 113^{3} + 52\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 77 + 68\cdot 113 + 105\cdot 113^{2} + 22\cdot 113^{3} + 66\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 104 + 14\cdot 113 + 100\cdot 113^{2} + 82\cdot 113^{3} + 14\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,5)(2,3,4,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
