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