Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 397 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 66 + 125\cdot 397 + 103\cdot 397^{2} + 244\cdot 397^{3} + 31\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 209 + 189\cdot 397 + 133\cdot 397^{2} + 339\cdot 397^{3} + 207\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 247 + 339\cdot 397 + 276\cdot 397^{2} + 291\cdot 397^{3} + 42\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 277 + 145\cdot 397 + 196\cdot 397^{2} + 258\cdot 397^{3} + 247\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 394 + 390\cdot 397 + 83\cdot 397^{2} + 57\cdot 397^{3} + 264\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
