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