Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 3\cdot 7 + 7^{2} + 2\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 4 + 4\cdot 7 + \left(5 a + 3\right)\cdot 7^{2} + \left(4 a + 4\right)\cdot 7^{3} + \left(4 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 + 6\cdot 7 + 7^{2} +O\left(7^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 5 + \left(6 a + 3\right)\cdot 7 + \left(a + 1\right)\cdot 7^{2} + \left(2 a + 4\right)\cdot 7^{3} + \left(2 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 6\cdot 7 + 4 a\cdot 7^{2} + \left(5 a + 4\right)\cdot 7^{3} + \left(2 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 2 + \left(6 a + 4\right)\cdot 7 + \left(2 a + 4\right)\cdot 7^{2} + \left(a + 5\right)\cdot 7^{3} + \left(4 a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $4$ | $3$ | $(3,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $-1$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
