Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 242\cdot 389 + 89\cdot 389^{2} + 11\cdot 389^{3} + 167\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 + 274\cdot 389 + 263\cdot 389^{2} + 360\cdot 389^{3} + 268\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 351 + 75\cdot 389 + 373\cdot 389^{2} + 124\cdot 389^{3} +O\left(389^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 372 + 185\cdot 389 + 51\cdot 389^{2} + 281\cdot 389^{3} + 341\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
