Basic invariants
Galois action
Roots of defining polynomial
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^{2} + 7 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 5 + a\cdot 11 + \left(9 a + 5\right)\cdot 11^{2} + 8 a\cdot 11^{3} + \left(a + 3\right)\cdot 11^{4} + \left(3 a + 7\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 4 + \left(7 a + 9\right)\cdot 11 + 10 a\cdot 11^{2} + \left(3 a + 6\right)\cdot 11^{3} + 3 a\cdot 11^{4} + \left(a + 10\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 5 + \left(7 a + 4\right)\cdot 11 + 8 a\cdot 11^{2} + 8\cdot 11^{3} + \left(3 a + 8\right)\cdot 11^{4} + \left(a + 2\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 10 + \left(9 a + 1\right)\cdot 11 + \left(a + 7\right)\cdot 11^{2} + \left(2 a + 4\right)\cdot 11^{3} + \left(9 a + 1\right)\cdot 11^{4} + \left(7 a + 7\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 2 + \left(3 a + 1\right)\cdot 11 + 3\cdot 11^{2} + 7 a\cdot 11^{3} + \left(7 a + 10\right)\cdot 11^{4} + 9 a\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 8 + \left(3 a + 4\right)\cdot 11 + \left(2 a + 5\right)\cdot 11^{2} + \left(10 a + 2\right)\cdot 11^{3} + \left(7 a + 9\right)\cdot 11^{4} + \left(9 a + 4\right)\cdot 11^{5} +O\left(11^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6,4,5,3)$ |
| $(1,4)(2,5)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
