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