Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 86\cdot 113 + 65\cdot 113^{2} + 112\cdot 113^{3} + 30\cdot 113^{4} + 18\cdot 113^{5} + 71\cdot 113^{6} + 12\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 15\cdot 113 + 28\cdot 113^{2} + 98\cdot 113^{3} + 106\cdot 113^{4} + 80\cdot 113^{5} + 60\cdot 113^{6} + 9\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 109\cdot 113 + 81\cdot 113^{2} + 92\cdot 113^{3} + 111\cdot 113^{4} + 34\cdot 113^{5} + 13\cdot 113^{6} + 111\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 46\cdot 113 + 82\cdot 113^{2} + 73\cdot 113^{3} + 34\cdot 113^{4} + 21\cdot 113^{5} + 46\cdot 113^{6} + 82\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 + 66\cdot 113 + 30\cdot 113^{2} + 39\cdot 113^{3} + 78\cdot 113^{4} + 91\cdot 113^{5} + 66\cdot 113^{6} + 30\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 3\cdot 113 + 31\cdot 113^{2} + 20\cdot 113^{3} + 113^{4} + 78\cdot 113^{5} + 99\cdot 113^{6} + 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 81 + 97\cdot 113 + 84\cdot 113^{2} + 14\cdot 113^{3} + 6\cdot 113^{4} + 32\cdot 113^{5} + 52\cdot 113^{6} + 103\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 112 + 26\cdot 113 + 47\cdot 113^{2} + 82\cdot 113^{4} + 94\cdot 113^{5} + 41\cdot 113^{6} + 100\cdot 113^{7} +O\left(113^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(2,7)(4,5)$ |
| $(1,2,3,5,8,7,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,2,3,5,8,7,6,4)$ | $0$ |
| $2$ | $8$ | $(1,5,6,2,8,4,3,7)$ | $0$ |
| $2$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $0$ |
| $2$ | $8$ | $(1,7,6,5,8,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
