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:
| $r_{ 1 }$ |
$=$ |
$ 5\cdot 7 + 6\cdot 7^{2} + 4\cdot 7^{3} + 7^{5} + 3\cdot 7^{6} +O\left(7^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + \left(6 a + 4\right)\cdot 7 + \left(5 a + 5\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} + \left(6 a + 5\right)\cdot 7^{5} + \left(5 a + 1\right)\cdot 7^{6} +O\left(7^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 6 + 3 a\cdot 7 + \left(5 a + 4\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(4 a + 2\right)\cdot 7^{4} + 5 a\cdot 7^{5} + \left(5 a + 4\right)\cdot 7^{6} +O\left(7^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 1 + 2\cdot 7 + \left(a + 5\right)\cdot 7^{2} + \left(a + 2\right)\cdot 7^{3} + \left(a + 1\right)\cdot 7^{4} + 6\cdot 7^{5} + a\cdot 7^{6} +O\left(7^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 1 + \left(3 a + 2\right)\cdot 7 + \left(a + 6\right)\cdot 7^{2} + 5\cdot 7^{3} + 2 a\cdot 7^{4} + \left(a + 1\right)\cdot 7^{5} + \left(a + 4\right)\cdot 7^{6} +O\left(7^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
