Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 + 7\cdot 269 + 177\cdot 269^{2} + 96\cdot 269^{3} + 5\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 70 + 54\cdot 269 + 147\cdot 269^{2} + 263\cdot 269^{3} + 182\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 81 + 160\cdot 269 + 117\cdot 269^{2} + 35\cdot 269^{3} + 214\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 143 + 101\cdot 269 + 187\cdot 269^{2} + 7\cdot 269^{3} + 217\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 212 + 214\cdot 269 + 177\cdot 269^{2} + 134\cdot 269^{3} + 187\cdot 269^{4} +O\left(269^{ 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.
