Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 20\cdot 113 + 5\cdot 113^{2} + 28\cdot 113^{3} + 34\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 78\cdot 113 + 55\cdot 113^{2} + 83\cdot 113^{3} + 88\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 + 98\cdot 113 + 51\cdot 113^{2} + 23\cdot 113^{3} + 65\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 96 + 2\cdot 113 + 102\cdot 113^{2} + 101\cdot 113^{3} + 19\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 111 + 25\cdot 113 + 11\cdot 113^{2} + 102\cdot 113^{3} + 17\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
