Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{3} + 2 x + 9 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a^{2} + 4 a + 4 + \left(8 a^{2} + 4 a + 2\right)\cdot 11 + \left(a + 9\right)\cdot 11^{2} + \left(5 a^{2} + 3 a\right)\cdot 11^{3} + \left(10 a^{2} + 2 a + 5\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a^{2} + 8 a + 5 + \left(4 a^{2} + 7 a + 5\right)\cdot 11 + \left(4 a^{2} + 5 a + 3\right)\cdot 11^{2} + \left(8 a^{2} + 8 a + 5\right)\cdot 11^{3} + \left(9 a^{2} + 6 a + 10\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a^{2} + 7 a + 4 + \left(6 a + 6\right)\cdot 11 + \left(6 a^{2} + 8 a + 9\right)\cdot 11^{2} + \left(4 a^{2} + 7 a + 6\right)\cdot 11^{3} + \left(3 a^{2} + 2 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a^{2} + 6 a + 1 + \left(7 a^{2} + a + 9\right)\cdot 11 + \left(7 a^{2} + 4 a + 3\right)\cdot 11^{2} + \left(10 a + 2\right)\cdot 11^{3} + \left(4 a^{2} + 6 a\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a^{2} + 2 a + 4 + \left(10 a^{2} + 4 a + 6\right)\cdot 11 + \left(5 a^{2} + 4 a + 5\right)\cdot 11^{2} + \left(3 a^{2} + 2 a + 2\right)\cdot 11^{3} + \left(4 a^{2} + 8 a + 3\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a^{2} + 10 a + 10 + \left(4 a^{2} + 4 a + 7\right)\cdot 11 + \left(9 a^{2} + 7 a + 6\right)\cdot 11^{2} + \left(5 a^{2} + 3 a + 8\right)\cdot 11^{3} + \left(5 a^{2} + 8 a + 7\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 10 a^{2} + a + 6 + \left(5 a^{2} + 5 a + 6\right)\cdot 11 + \left(2 a^{2} + 5 a\right)\cdot 11^{2} + \left(5 a^{2} + 8 a + 1\right)\cdot 11^{3} + \left(7 a^{2} + a + 1\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 3 a^{2} + 5 a + 2 + \left(5 a^{2} + 10 a + 1\right)\cdot 11 + \left(6 a^{2} + 5 a + 10\right)\cdot 11^{2} + \left(10 a + 8\right)\cdot 11^{3} + \left(2 a^{2} + 10 a + 6\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 5 a^{2} + a + 9 + \left(7 a^{2} + 10 a + 9\right)\cdot 11 + 5\cdot 11^{2} + \left(10 a^{2} + 7\right)\cdot 11^{3} + \left(7 a^{2} + 7 a\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,6,7,2,3,4,9,8)$ |
| $(1,7,4)(2,9,5)(3,8,6)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,3)(2,5)(4,8)(6,7)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,7,4)(2,9,5)(3,8,6)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,5,6,7,2,3,4,9,8)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,6,2,4,8,5,7,3,9)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,2,8,7,9,6,4,5,3)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
