Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 367 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 65 + 54\cdot 367 + 318\cdot 367^{2} + 348\cdot 367^{3} + 228\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 119 + 328\cdot 367 + 350\cdot 367^{2} + 32\cdot 367^{3} + 76\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 201 + 68\cdot 367 + 186\cdot 367^{2} + 65\cdot 367^{3} + 105\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 349 + 282\cdot 367 + 245\cdot 367^{2} + 286\cdot 367^{3} + 323\cdot 367^{4} +O\left(367^{ 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.
