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:
| $r_{ 1 }$ |
$=$ |
$ 9 + 6\cdot 17 + 16\cdot 17^{2} + 3\cdot 17^{3} + 9\cdot 17^{4} + 5\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a^{2} + 6 a + 5 + \left(a^{2} + 6 a + 9\right)\cdot 17 + \left(4 a^{2} + 2 a + 6\right)\cdot 17^{2} + \left(12 a^{2} + 14 a + 15\right)\cdot 17^{3} + \left(2 a^{2} + 12 a + 11\right)\cdot 17^{4} + \left(7 a^{2} + 7 a + 12\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a^{2} + 5 a + \left(16 a^{2} + 3 a + 14\right)\cdot 17 + \left(a^{2} + a + 14\right)\cdot 17^{2} + 8 a\cdot 17^{3} + \left(2 a^{2} + 10 a + 9\right)\cdot 17^{4} + \left(10 a^{2} + 8 a + 10\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 8\cdot 17 + 16\cdot 17^{2} + 5\cdot 17^{3} + 5\cdot 17^{4} + 10\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a^{2} + a + 6 + \left(10 a^{2} + 12 a + 10\right)\cdot 17 + \left(2 a^{2} + 15 a + 9\right)\cdot 17^{2} + \left(14 a^{2} + 15 a + 4\right)\cdot 17^{3} + \left(5 a^{2} + 6 a\right)\cdot 17^{4} + \left(6 a^{2} + 2 a + 8\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a^{2} + a + \left(14 a^{2} + a + 1\right)\cdot 17 + \left(14 a^{2} + 15 a + 8\right)\cdot 17^{2} + \left(6 a^{2} + 2 a\right)\cdot 17^{3} + \left(3 a^{2} + 11 a + 1\right)\cdot 17^{4} + \left(8 a^{2} + 12 a + 2\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 a^{2} + 11 a + 9 + \left(7 a^{2} + a + 2\right)\cdot 17 + \left(12 a^{2} + 16\right)\cdot 17^{2} + \left(2 a^{2} + 10 a + 13\right)\cdot 17^{3} + \left(9 a^{2} + 16 a + 13\right)\cdot 17^{4} + \left(5 a + 9\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 a^{2} + 10 a + 8 + \left(a^{2} + 9 a + 15\right)\cdot 17 + \left(15 a^{2} + 16 a + 13\right)\cdot 17^{2} + \left(14 a^{2} + 16 a + 5\right)\cdot 17^{3} + \left(10 a^{2} + 9 a\right)\cdot 17^{4} + \left(a^{2} + 13 a + 9\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(3,7)(4,5)$ |
| $(1,8,2)(3,4,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-3$ |
| $3$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $-1$ |
| $3$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $1$ |
| $4$ | $3$ | $(1,2,8)(3,5,4)$ | $0$ |
| $4$ | $3$ | $(1,8,2)(3,4,5)$ | $0$ |
| $4$ | $6$ | $(1,3,8,4,2,5)(6,7)$ | $0$ |
| $4$ | $6$ | $(1,5,2,4,8,3)(6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
