Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 9\cdot 11^{2} + 11^{3} + 10\cdot 11^{4} + 8\cdot 11^{5} + 5\cdot 11^{6} +O\left(11^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 3\cdot 11 + 8\cdot 11^{2} + 5\cdot 11^{3} + 2\cdot 11^{4} + 2\cdot 11^{5} + 2\cdot 11^{6} +O\left(11^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 7\cdot 11 + 2\cdot 11^{2} + 5\cdot 11^{3} + 8\cdot 11^{4} + 8\cdot 11^{5} + 8\cdot 11^{6} +O\left(11^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 10\cdot 11 + 11^{2} + 9\cdot 11^{3} + 2\cdot 11^{5} + 5\cdot 11^{6} +O\left(11^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,3,4,2)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
