Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 154\cdot 277 + 271\cdot 277^{2} + 170\cdot 277^{3} + 206\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 145 + 54\cdot 277 + 47\cdot 277^{2} + 208\cdot 277^{3} + 90\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 184 + 269\cdot 277 + 242\cdot 277^{2} + 209\cdot 277^{3} + 31\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 228 + 192\cdot 277 + 114\cdot 277^{2} + 122\cdot 277^{3} + 272\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 271 + 159\cdot 277 + 154\cdot 277^{2} + 119\cdot 277^{3} + 229\cdot 277^{4} +O\left(277^{ 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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
