Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 307 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 77 + 196\cdot 307 + 252\cdot 307^{2} + 141\cdot 307^{3} + 285\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 137 + 232\cdot 307 + 37\cdot 307^{2} + 65\cdot 307^{3} + 143\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 182 + 114\cdot 307 + 254\cdot 307^{2} + 30\cdot 307^{3} + 184\cdot 307^{4} +O\left(307^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 220 + 70\cdot 307 + 69\cdot 307^{2} + 69\cdot 307^{3} + 307^{4} +O\left(307^{ 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.
