Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{2} + 7 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 3\cdot 11 + 5\cdot 11^{2} + 3\cdot 11^{3} + 10\cdot 11^{4} + 8\cdot 11^{6} + 2\cdot 11^{7} + 6\cdot 11^{8} +O\left(11^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 8 + \left(7 a + 9\right)\cdot 11 + \left(2 a + 2\right)\cdot 11^{2} + \left(10 a + 9\right)\cdot 11^{3} + \left(5 a + 5\right)\cdot 11^{4} + 6\cdot 11^{5} + \left(4 a + 9\right)\cdot 11^{6} + \left(8 a + 2\right)\cdot 11^{7} + \left(5 a + 2\right)\cdot 11^{8} +O\left(11^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 4\cdot 11 + 3\cdot 11^{2} + 9\cdot 11^{3} + 10\cdot 11^{4} + 4\cdot 11^{5} + 7\cdot 11^{6} + 2\cdot 11^{7} + 11^{8} +O\left(11^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 4 + 9 a\cdot 11 + \left(4 a + 6\right)\cdot 11^{2} + \left(2 a + 1\right)\cdot 11^{3} + \left(2 a + 1\right)\cdot 11^{4} + \left(a + 10\right)\cdot 11^{5} + \left(4 a + 4\right)\cdot 11^{6} + \left(9 a + 9\right)\cdot 11^{7} + 6 a\cdot 11^{8} +O\left(11^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 2 + 5 a\cdot 11 + \left(a + 2\right)\cdot 11^{2} + \left(3 a + 5\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} + \left(3 a + 5\right)\cdot 11^{5} + 9 a\cdot 11^{6} + \left(10 a + 8\right)\cdot 11^{7} + \left(7 a + 4\right)\cdot 11^{8} +O\left(11^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 8 + \left(5 a + 4\right)\cdot 11 + \left(9 a + 2\right)\cdot 11^{2} + \left(7 a + 5\right)\cdot 11^{3} + 2\cdot 11^{4} + 7 a\cdot 11^{5} + \left(a + 1\right)\cdot 11^{6} + 9\cdot 11^{7} + \left(3 a + 3\right)\cdot 11^{8} +O\left(11^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 4 a + 3 + 3 a\cdot 11 + \left(8 a + 6\right)\cdot 11^{2} + 3\cdot 11^{3} + \left(5 a + 8\right)\cdot 11^{4} + \left(10 a + 2\right)\cdot 11^{5} + \left(6 a + 3\right)\cdot 11^{6} + \left(2 a + 10\right)\cdot 11^{7} + \left(5 a + 5\right)\cdot 11^{8} +O\left(11^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 5 a + 6 + \left(a + 10\right)\cdot 11 + \left(6 a + 4\right)\cdot 11^{2} + \left(8 a + 6\right)\cdot 11^{3} + \left(8 a + 7\right)\cdot 11^{4} + \left(9 a + 1\right)\cdot 11^{5} + \left(6 a + 9\right)\cdot 11^{6} + \left(a + 9\right)\cdot 11^{7} + \left(4 a + 7\right)\cdot 11^{8} +O\left(11^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,4)(5,7,6,8)$ |
| $(2,8,6)(4,7,5)$ |
| $(1,7,3,8)(2,5,4,6)$ |
| $(2,4)(5,8)(6,7)$ |
| $(1,3)(2,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $-2$ |
| $12$ | $2$ | $(2,4)(5,8)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,2,5)(3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,2,3,4)(5,7,6,8)$ | $0$ |
| $8$ | $6$ | $(1,6,2,3,5,4)(7,8)$ | $1$ |
| $6$ | $8$ | $(1,7,2,6,3,8,4,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,8,2,5,3,7,4,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
