Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 90 + 167\cdot 389 + 17\cdot 389^{2} + 311\cdot 389^{3} + 244\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 225 + 260\cdot 389 + 182\cdot 389^{2} + 139\cdot 389^{3} + 112\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 239 + 194\cdot 389 + 168\cdot 389^{2} + 65\cdot 389^{3} + 181\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 300 + 378\cdot 389 + 265\cdot 389^{2} + 301\cdot 389^{3} + 247\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 313 + 165\cdot 389 + 143\cdot 389^{2} + 349\cdot 389^{3} + 380\cdot 389^{4} +O\left(389^{ 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 |
| $1$ | $1$ | $()$ | $4$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
