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