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