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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 8\cdot 11 + 7\cdot 11^{2} + 9\cdot 11^{3} + 3\cdot 11^{4} + 8\cdot 11^{5} + 4\cdot 11^{6} + 2\cdot 11^{7} + 9\cdot 11^{8} + 10\cdot 11^{10} +O\left(11^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a^{2} + 7 a + 6 + \left(5 a^{2} + 8 a + 4\right)\cdot 11 + \left(5 a^{2} + 4 a + 9\right)\cdot 11^{2} + \left(2 a^{2} + 5 a + 2\right)\cdot 11^{3} + \left(7 a + 6\right)\cdot 11^{4} + \left(a^{2} + 9\right)\cdot 11^{5} + \left(8 a^{2} + 8 a + 8\right)\cdot 11^{6} + \left(2 a^{2} + 9 a + 2\right)\cdot 11^{7} + \left(10 a^{2} + 7 a + 6\right)\cdot 11^{8} + \left(6 a^{2} + 8 a + 10\right)\cdot 11^{9} + \left(7 a^{2} + a + 7\right)\cdot 11^{10} +O\left(11^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a^{2} + 4 a + 4 + \left(3 a^{2} + 2\right)\cdot 11 + \left(2 a^{2} + 4 a + 5\right)\cdot 11^{2} + \left(a^{2} + 8 a + 8\right)\cdot 11^{3} + \left(8 a^{2} + 7 a + 5\right)\cdot 11^{4} + \left(6 a^{2} + a + 2\right)\cdot 11^{5} + 6\cdot 11^{6} + \left(6 a^{2} + a + 3\right)\cdot 11^{7} + \left(a^{2} + 5 a + 9\right)\cdot 11^{8} + \left(3 a^{2} + 3 a + 1\right)\cdot 11^{9} + \left(9 a^{2} + 2 a + 10\right)\cdot 11^{10} +O\left(11^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a^{2} + 6 + \left(a^{2} + 2 a + 6\right)\cdot 11 + \left(3 a^{2} + 2 a + 2\right)\cdot 11^{2} + \left(7 a^{2} + 8 a + 9\right)\cdot 11^{3} + \left(2 a^{2} + 6 a + 5\right)\cdot 11^{4} + \left(3 a^{2} + 8 a + 1\right)\cdot 11^{5} + \left(2 a^{2} + 2 a + 1\right)\cdot 11^{6} + \left(2 a^{2} + 2\right)\cdot 11^{7} + \left(10 a^{2} + 9 a + 6\right)\cdot 11^{8} + \left(9 a + 2\right)\cdot 11^{9} + \left(5 a^{2} + 6 a + 8\right)\cdot 11^{10} +O\left(11^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a^{2} + 10 a + 7 + \left(a^{2} + 3 a + 5\right)\cdot 11 + \left(4 a^{2} + 2 a + 6\right)\cdot 11^{2} + \left(9 a^{2} + 4 a\right)\cdot 11^{3} + \left(10 a^{2} + 3 a + 5\right)\cdot 11^{4} + \left(10 a^{2} + 8 a + 9\right)\cdot 11^{5} + \left(a^{2} + 4 a + 6\right)\cdot 11^{6} + \left(a^{2} + 2 a\right)\cdot 11^{7} + \left(2 a^{2} + 4 a + 2\right)\cdot 11^{8} + \left(7 a^{2} + a + 10\right)\cdot 11^{9} + \left(10 a^{2} + 10 a + 8\right)\cdot 11^{10} +O\left(11^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a^{2} + 7 a + 1 + \left(9 a^{2} + 10 a + 9\right)\cdot 11 + \left(5 a^{2} + 10 a + 8\right)\cdot 11^{2} + \left(2 a^{2} + 5 a + 9\right)\cdot 11^{3} + \left(4 a^{2} + 2 a + 10\right)\cdot 11^{4} + \left(7 a^{2} + 5 a\right)\cdot 11^{5} + \left(8 a^{2} + 9 a + 1\right)\cdot 11^{6} + \left(2 a^{2} + 10 a + 10\right)\cdot 11^{7} + \left(5 a^{2} + 10 a + 9\right)\cdot 11^{8} + \left(6 a^{2} + 2 a + 1\right)\cdot 11^{9} + \left(8 a^{2} + 9 a + 6\right)\cdot 11^{10} +O\left(11^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 + 11 + 5\cdot 11^{2} + 5\cdot 11^{3} + 6\cdot 11^{4} + 4\cdot 11^{5} + 10\cdot 11^{6} + 2\cdot 11^{7} + 4\cdot 11^{8} + 4\cdot 11^{9} + 11^{10} +O\left(11^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 3 a^{2} + 5 a + 10 + \left(10 a^{2} + 7 a + 5\right)\cdot 11 + \left(8 a + 9\right)\cdot 11^{2} + \left(10 a^{2} + 8\right)\cdot 11^{3} + \left(6 a^{2} + 5 a + 10\right)\cdot 11^{4} + \left(3 a^{2} + 8 a + 6\right)\cdot 11^{5} + \left(7 a + 4\right)\cdot 11^{6} + \left(7 a^{2} + 8 a + 8\right)\cdot 11^{7} + \left(3 a^{2} + 6 a + 7\right)\cdot 11^{8} + \left(8 a^{2} + 6 a\right)\cdot 11^{9} + \left(2 a^{2} + 2 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 |
| $(1,3,8)(2,7,5)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
| $(1,6,7,4)(2,3,8,5)$ |
| $(1,2,7,8)(3,6,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $4$ | $3$ | $(2,4,3)(5,8,6)$ | $\zeta_{3} + 1$ |
| $4$ | $3$ | $(2,3,4)(5,6,8)$ | $-\zeta_{3}$ |
| $6$ | $4$ | $(1,2,7,8)(3,6,5,4)$ | $0$ |
| $4$ | $6$ | $(1,7)(2,5,4,8,3,6)$ | $\zeta_{3}$ |
| $4$ | $6$ | $(1,7)(2,6,3,8,4,5)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
