Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 349 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 47 + 249\cdot 349 + 190\cdot 349^{2} + 89\cdot 349^{3} + 228\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 106 + 124\cdot 349 + 205\cdot 349^{2} + 296\cdot 349^{3} + 191\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 116 + 333\cdot 349 + 319\cdot 349^{2} + 166\cdot 349^{3} + 43\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 181 + 168\cdot 349 + 25\cdot 349^{2} + 178\cdot 349^{3} + 226\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 250 + 171\cdot 349 + 305\cdot 349^{2} + 315\cdot 349^{3} + 7\cdot 349^{4} +O\left(349^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$-2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
