Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 313 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 94 + 32\cdot 313 + 286\cdot 313^{2} + 301\cdot 313^{3} + 282\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 96 + 141\cdot 313 + 90\cdot 313^{2} + 237\cdot 313^{3} + 125\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 191 + 107\cdot 313 + 212\cdot 313^{2} + 5\cdot 313^{3} + 104\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 245 + 31\cdot 313 + 37\cdot 313^{2} + 81\cdot 313^{3} + 113\cdot 313^{4} +O\left(313^{ 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.
