Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 5 + \left(5 a + 3\right)\cdot 7 + \left(5 a + 3\right)\cdot 7^{2} + \left(a + 2\right)\cdot 7^{3} + 5 a\cdot 7^{4} + 2\cdot 7^{5} + 6 a\cdot 7^{6} + \left(2 a + 5\right)\cdot 7^{7} + \left(6 a + 2\right)\cdot 7^{8} +O\left(7^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 5 + \left(6 a + 5\right)\cdot 7 + 3\cdot 7^{2} + \left(2 a + 2\right)\cdot 7^{3} + \left(4 a + 2\right)\cdot 7^{4} + \left(6 a + 1\right)\cdot 7^{5} + \left(3 a + 6\right)\cdot 7^{6} + \left(6 a + 6\right)\cdot 7^{7} + \left(4 a + 5\right)\cdot 7^{8} +O\left(7^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 1 + \left(a + 6\right)\cdot 7 + \left(a + 3\right)\cdot 7^{2} + \left(5 a + 5\right)\cdot 7^{3} + \left(a + 3\right)\cdot 7^{4} + \left(6 a + 4\right)\cdot 7^{5} + 5\cdot 7^{6} + \left(4 a + 1\right)\cdot 7^{7} + 6\cdot 7^{8} +O\left(7^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + \left(4 a + 5\right)\cdot 7 + \left(a + 3\right)\cdot 7^{2} + 3 a\cdot 7^{4} + \left(4 a + 4\right)\cdot 7^{5} + \left(4 a + 2\right)\cdot 7^{6} + \left(3 a + 6\right)\cdot 7^{7} + \left(a + 1\right)\cdot 7^{8} +O\left(7^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 6 + 3\cdot 7 + \left(6 a + 5\right)\cdot 7^{2} + \left(4 a + 3\right)\cdot 7^{3} + \left(2 a + 4\right)\cdot 7^{4} + 3\cdot 7^{5} + \left(3 a + 3\right)\cdot 7^{6} + 2\cdot 7^{7} + \left(2 a + 4\right)\cdot 7^{8} +O\left(7^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 6 + \left(2 a + 3\right)\cdot 7 + 5 a\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} + \left(2 a + 5\right)\cdot 7^{5} + \left(2 a + 2\right)\cdot 7^{6} + \left(3 a + 5\right)\cdot 7^{7} + \left(5 a + 6\right)\cdot 7^{8} +O\left(7^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6,3)$ |
| $(1,3,5,2,4,6)$ |
| $(1,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,6,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,3,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,4)(2,3,6)$ | $-1$ |
| $3$ | $6$ | $(1,3,5,2,4,6)$ | $0$ |
| $3$ | $6$ | $(1,6,4,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
