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 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.
