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