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