Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 37\cdot 113 + 11\cdot 113^{2} + 72\cdot 113^{3} + 106\cdot 113^{4} + 68\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 103\cdot 113 + 39\cdot 113^{2} + 6\cdot 113^{3} + 17\cdot 113^{4} + 35\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 19\cdot 113 + 104\cdot 113^{2} + 7\cdot 113^{3} + 92\cdot 113^{4} + 2\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 66\cdot 113 + 70\cdot 113^{2} + 26\cdot 113^{3} + 10\cdot 113^{4} + 6\cdot 113^{5} +O\left(113^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
