Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 331 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 170\cdot 331 + 73\cdot 331^{2} + 152\cdot 331^{3} + 327\cdot 331^{4} +O\left(331^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 80 + 255\cdot 331 + 198\cdot 331^{2} + 329\cdot 331^{3} + 50\cdot 331^{4} +O\left(331^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 245 + 181\cdot 331 + 289\cdot 331^{2} + 236\cdot 331^{3} + 245\cdot 331^{4} +O\left(331^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 308 + 54\cdot 331 + 100\cdot 331^{2} + 274\cdot 331^{3} + 37\cdot 331^{4} +O\left(331^{ 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.
