Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{3} + x + 14 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a^{2} + 8 a + 14 + \left(4 a^{2} + 5\right)\cdot 17 + \left(8 a^{2} + 9 a + 11\right)\cdot 17^{2} + \left(6 a^{2} + 11\right)\cdot 17^{3} + \left(7 a^{2} + a + 3\right)\cdot 17^{4} + a^{2}17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a^{2} + 3 a + 4 + \left(15 a^{2} + 13\right)\cdot 17 + \left(7 a^{2} + 16\right)\cdot 17^{2} + \left(8 a + 1\right)\cdot 17^{3} + \left(3 a^{2} + 4 a + 12\right)\cdot 17^{4} + \left(15 a^{2} + 2 a + 3\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a^{2} + 7 a + 8 + \left(9 a + 13\right)\cdot 17 + \left(7 a^{2} + 8 a + 3\right)\cdot 17^{2} + \left(12 a^{2} + 8 a + 5\right)\cdot 17^{3} + \left(2 a^{2} + 10 a + 10\right)\cdot 17^{4} + \left(16 a^{2} + 7 a + 9\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a^{2} + 6 a + \left(13 a^{2} + 16 a + 6\right)\cdot 17 + \left(7 a + 6\right)\cdot 17^{2} + \left(10 a^{2} + 8 a + 8\right)\cdot 17^{3} + \left(6 a^{2} + 11 a + 14\right)\cdot 17^{4} + \left(14 a + 10\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 3\cdot 17 + 2\cdot 17^{2} + 4\cdot 17^{3} + 12\cdot 17^{4} + 5\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a^{2} + 10 a + 15 + \left(11 a^{2} + a + 3\right)\cdot 17 + \left(2 a^{2} + 9 a + 12\right)\cdot 17^{2} + \left(8 a^{2} + 9 a + 13\right)\cdot 17^{3} + \left(2 a^{2} + a + 15\right)\cdot 17^{4} + \left(11 a^{2} + 3 a + 11\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ a^{2} + 4 + \left(5 a^{2} + 6 a + 5\right)\cdot 17 + \left(7 a^{2} + 16 a + 15\right)\cdot 17^{2} + \left(13 a^{2} + 15 a + 5\right)\cdot 17^{3} + \left(11 a^{2} + 4 a + 16\right)\cdot 17^{4} + \left(6 a^{2} + 6 a + 8\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(2,3,7,5)(4,6)$ |
| $(1,5)(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$7$ |
| $21$ |
$2$ |
$(1,5)(2,4)$ |
$-1$ |
| $56$ |
$3$ |
$(1,2,7)(3,5,6)$ |
$1$ |
| $42$ |
$4$ |
$(2,3,7,5)(4,6)$ |
$-1$ |
| $24$ |
$7$ |
$(1,5,4,6,2,3,7)$ |
$0$ |
| $24$ |
$7$ |
$(1,6,7,4,3,5,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
