Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 6 + a\cdot 7 + \left(a + 3\right)\cdot 7^{2} + \left(4 a + 4\right)\cdot 7^{3} + \left(4 a + 3\right)\cdot 7^{4} + \left(4 a + 4\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 3 + \left(5 a + 3\right)\cdot 7 + 6\cdot 7^{2} + 6 a\cdot 7^{3} + \left(6 a + 3\right)\cdot 7^{4} + \left(4 a + 5\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 1 + 5 a\cdot 7 + \left(5 a + 3\right)\cdot 7^{2} + 2 a\cdot 7^{3} + \left(2 a + 4\right)\cdot 7^{4} + \left(2 a + 4\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 + 3\cdot 7 + 6\cdot 7^{2} + 7^{3} + 6\cdot 7^{4} + 2\cdot 7^{5} +O\left(7^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 6 + \left(a + 5\right)\cdot 7 + \left(6 a + 1\right)\cdot 7^{2} + 6\cdot 7^{3} + 3\cdot 7^{4} + \left(2 a + 3\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,3)(2,5)$ |
| $(1,2)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $5$ |
$2$ |
$(1,3)(2,5)$ |
$0$ |
$0$ |
| $2$ |
$5$ |
$(1,5,4,2,3)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
| $2$ |
$5$ |
$(1,4,3,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$\zeta_{5}^{3} + \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
