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