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