Basic invariants
Defining polynomial
| $f(x)$ | $=$ | $ x^{9} - x^{8} - 16 x^{7} + 11 x^{6} + 66 x^{5} - 32 x^{4} - 73 x^{3} + 7 x^{2} + 7 x - 1 $. |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{3} + 2 x + 9 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a^{2} + 8 a + 2 + \left(5 a + 5\right)\cdot 11 + \left(6 a^{2} + 3 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 5 a + 2\right)\cdot 11^{3} + \left(10 a^{2} + 9 a + 5\right)\cdot 11^{4} + \left(3 a^{2} + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a^{2} + 2 a + 6 + \left(9 a^{2} + a + 6\right)\cdot 11 + \left(a^{2} + a + 6\right)\cdot 11^{2} + \left(7 a^{2} + 10 a + 10\right)\cdot 11^{3} + \left(a^{2} + a + 3\right)\cdot 11^{4} + \left(6 a^{2} + 8 a + 9\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + \left(6 a^{2} + 5 a + 4\right)\cdot 11 + \left(9 a + 4\right)\cdot 11^{2} + \left(9 a^{2} + a + 4\right)\cdot 11^{3} + \left(9 a + 4\right)\cdot 11^{4} + \left(10 a^{2} + 3 a + 8\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a^{2} + 2 a + 5 + \left(3 a^{2} + 7 a + 7\right)\cdot 11 + \left(7 a^{2} + 2 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(5 a^{2} + 4 a + 3\right)\cdot 11^{4} + \left(3 a^{2} + a + 3\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a^{2} + 4 a + 3 + \left(7 a^{2} + 5 a + 9\right)\cdot 11 + \left(3 a^{2} + a\right)\cdot 11^{2} + \left(2 a^{2} + a + 6\right)\cdot 11^{3} + \left(4 a^{2} + a\right)\cdot 11^{4} + \left(6 a^{2} + 2 a + 5\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a^{2} + 10 a + 6 + \left(a^{2} + 8 a + 9\right)\cdot 11 + \left(3 a^{2} + 9 a + 7\right)\cdot 11^{2} + \left(6 a^{2} + a\right)\cdot 11^{3} + \left(4 a^{2} + 8 a + 2\right)\cdot 11^{4} + \left(8 a^{2} + 5 a + 6\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 9 a^{2} + 6 a + \left(10 a^{2} + 5 a + 3\right)\cdot 11 + \left(5 a^{2} + a\right)\cdot 11^{2} + \left(3 a^{2} + 8 a + 4\right)\cdot 11^{3} + \left(7 a^{2} + 4 a + 8\right)\cdot 11^{4} + \left(5 a^{2} + 7 a + 7\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 10 a^{2} + a + 5 + \left(3 a^{2} + 8\right)\cdot 11 + \left(a^{2} + 8 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + a + 9\right)\cdot 11^{3} + \left(10 a^{2} + 5 a + 8\right)\cdot 11^{4} + \left(9 a^{2} + a + 9\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 7 a^{2} + a + 7 + \left(4 a + 1\right)\cdot 11 + \left(3 a^{2} + 6 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + 6 a\right)\cdot 11^{3} + \left(9 a^{2} + 10 a + 7\right)\cdot 11^{4} + \left(a + 9\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6,8,2,3,7,9,4,5)$ |
| $(1,9,2)(3,6,4)(5,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,2,9)(3,4,6)(5,8,7)$ | $\zeta_{9}^{3}$ |
| $1$ | $3$ | $(1,9,2)(3,6,4)(5,7,8)$ | $-\zeta_{9}^{3} - 1$ |
| $1$ | $9$ | $(1,6,8,2,3,7,9,4,5)$ | $\zeta_{9}$ |
| $1$ | $9$ | $(1,8,3,9,5,6,2,7,4)$ | $\zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,3,5,2,4,8,9,6,7)$ | $\zeta_{9}^{4}$ |
| $1$ | $9$ | $(1,7,6,9,8,4,2,5,3)$ | $\zeta_{9}^{5}$ |
| $1$ | $9$ | $(1,4,7,2,6,5,9,3,8)$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ | $9$ | $(1,5,4,9,7,3,2,8,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
