Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 3\cdot 11 + 7\cdot 11^{3} + 2\cdot 11^{4} + 3\cdot 11^{5} + 4\cdot 11^{6} + 8\cdot 11^{7} +O\left(11^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 4\cdot 11 + 10\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} + 10\cdot 11^{5} + 3\cdot 11^{6} + 10\cdot 11^{7} +O\left(11^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 6\cdot 11 + 11^{3} + 9\cdot 11^{4} + 7\cdot 11^{6} +O\left(11^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 7\cdot 11 + 10\cdot 11^{2} + 3\cdot 11^{3} + 8\cdot 11^{4} + 7\cdot 11^{5} + 6\cdot 11^{6} + 2\cdot 11^{7} +O\left(11^{ 8 }\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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,3,4,2)$ |
$\zeta_{4}$ |
$-\zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,4,3)$ |
$-\zeta_{4}$ |
$\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
