Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 326\cdot 379 + 277\cdot 379^{2} + 46\cdot 379^{3} + 171\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 113 + 344\cdot 379 + 308\cdot 379^{2} + 194\cdot 379^{3} + 61\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 146 + 289\cdot 379 + 18\cdot 379^{2} + 59\cdot 379^{3} + 333\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 246 + 145\cdot 379 + 96\cdot 379^{2} + 261\cdot 379^{3} + 246\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 248 + 31\cdot 379 + 56\cdot 379^{2} + 196\cdot 379^{3} + 324\cdot 379^{4} +O\left(379^{ 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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
