Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 38\cdot 269 + 220\cdot 269^{2} + 266\cdot 269^{3} + 205\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 234\cdot 269 + 195\cdot 269^{2} + 160\cdot 269^{3} + 264\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 211 + 93\cdot 269 + 159\cdot 269^{2} + 5\cdot 269^{3} + 232\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 260 + 228\cdot 269 + 179\cdot 269^{2} + 112\cdot 269^{3} + 219\cdot 269^{4} +O\left(269^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 262 + 211\cdot 269 + 51\cdot 269^{2} + 261\cdot 269^{3} + 153\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.
