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^{3} + 6 x^{2} + 4 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a^{2} + 3 a + \left(4 a^{2} + 2 a\right)\cdot 7 + \left(a^{2} + 5 a + 3\right)\cdot 7^{2} + \left(3 a^{2} + a + 4\right)\cdot 7^{3} + \left(2 a^{2} + 6 a\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a^{2} + 4 a + 6 + \left(a^{2} + 4 a + 1\right)\cdot 7 + \left(a + 5\right)\cdot 7^{2} + \left(2 a^{2} + 2 a + 5\right)\cdot 7^{3} + \left(2 a^{2} + 4 a + 5\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a^{2} + 4 a + 3 + \left(a^{2} + a + 5\right)\cdot 7 + \left(3 a^{2} + 6 a + 1\right)\cdot 7^{2} + \left(4 a^{2} + 2 a + 1\right)\cdot 7^{3} + \left(4 a^{2} + 2 a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a^{2} + 5 + \left(a^{2} + 6\right)\cdot 7 + \left(5 a^{2} + 4\right)\cdot 7^{2} + \left(a^{2} + 3 a + 3\right)\cdot 7^{3} + \left(2 a^{2} + 3 a + 4\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a^{2} + 2 a + 1 + \left(2 a^{2} + 3\right)\cdot 7 + \left(4 a^{2} + 2 a\right)\cdot 7^{2} + \left(3 a^{2} + 2 a + 3\right)\cdot 7^{3} + \left(a^{2} + 3 a + 3\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a^{2} + a + \left(2 a^{2} + 5 a + 4\right)\cdot 7 + \left(6 a^{2} + 5 a + 5\right)\cdot 7^{2} + \left(5 a^{2} + a + 2\right)\cdot 7^{3} + a\cdot 7^{4} +O\left(7^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,5)(2,6)(3,4)$ | $2$ |
| $15$ | $2$ | $(2,4)(3,6)$ | $0$ |
| $20$ | $3$ | $(1,5,3)(2,6,4)$ | $1$ |
| $30$ | $4$ | $(2,5,3,4)$ | $0$ |
| $24$ | $5$ | $(1,6,3,5,2)$ | $-1$ |
| $20$ | $6$ | $(1,6,5,4,3,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
