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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 1 + \left(6 a + 7\right)\cdot 11 + \left(8 a + 5\right)\cdot 11^{2} + \left(10 a + 5\right)\cdot 11^{3} + a\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 7 + \left(4 a + 2\right)\cdot 11 + 3 a\cdot 11^{2} + \left(5 a + 9\right)\cdot 11^{3} + \left(4 a + 3\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + \left(2 a + 6\right)\cdot 11 + \left(3 a + 1\right)\cdot 11^{2} + \left(4 a + 3\right)\cdot 11^{3} + 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 9 + \left(8 a + 10\right)\cdot 11 + 7 a\cdot 11^{2} + \left(6 a + 6\right)\cdot 11^{3} + \left(10 a + 9\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 7 + \left(4 a + 4\right)\cdot 11 + 2 a\cdot 11^{2} + 7\cdot 11^{3} + \left(9 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 3 + \left(6 a + 1\right)\cdot 11 + \left(7 a + 9\right)\cdot 11^{2} + \left(5 a + 4\right)\cdot 11^{3} + \left(6 a + 5\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 + 4\cdot 11^{2} + 8\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,3)(4,7)(5,6)$ |
| $(1,2)(3,7)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $7$ |
$2$ |
$(1,3)(4,7)(5,6)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$7$ |
$(1,7,5,6,4,3,2)$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ |
$7$ |
$(1,5,4,2,7,6,3)$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ |
$7$ |
$(1,6,2,5,3,7,4)$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
