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