Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{3} + x + 14 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a^{2} + 11 a + 2 + \left(7 a^{2} + 8 a + 6\right)\cdot 17 + \left(3 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(2 a^{2} + 7 a\right)\cdot 17^{3} + \left(4 a^{2} + 16\right)\cdot 17^{4} + \left(10 a^{2} + 6 a + 11\right)\cdot 17^{5} + \left(2 a^{2} + 12 a + 11\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a^{2} + 15 a + \left(14 a^{2} + 9 a + 3\right)\cdot 17 + \left(6 a^{2} + 14 a + 12\right)\cdot 17^{2} + \left(14 a^{2} + 13 a + 10\right)\cdot 17^{3} + \left(7 a^{2} + 12 a + 15\right)\cdot 17^{4} + \left(6 a^{2} + a + 15\right)\cdot 17^{5} + \left(a^{2} + 6 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a^{2} + 8 a + 9 + \left(5 a^{2} + 3 a + 9\right)\cdot 17 + \left(9 a^{2} + 10 a + 11\right)\cdot 17^{2} + \left(15 a^{2} + 10 a + 4\right)\cdot 17^{3} + \left(a^{2} + 11 a\right)\cdot 17^{4} + \left(13 a^{2} + 11 a + 9\right)\cdot 17^{5} + \left(10 a^{2} + 8 a + 6\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a^{2} + 13 a + 12 + \left(8 a^{2} + 11 a + 5\right)\cdot 17 + \left(4 a^{2} + 11 a + 8\right)\cdot 17^{2} + \left(14 a^{2} + 5 a + 9\right)\cdot 17^{3} + \left(2 a^{2} + 10 a + 6\right)\cdot 17^{4} + \left(10 a^{2} + 6 a + 1\right)\cdot 17^{5} + \left(7 a^{2} + 11 a + 10\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a^{2} + 10 a + \left(2 a + 5\right)\cdot 17 + \left(11 a^{2} + 7 a + 9\right)\cdot 17^{2} + \left(11 a^{2} + 5 a + 14\right)\cdot 17^{3} + \left(14 a^{2} + 10 a + 8\right)\cdot 17^{4} + \left(5 a^{2} + 10 a + 15\right)\cdot 17^{5} + \left(8 a^{2} + 3 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a^{2} + 5 a + 10 + \left(10 a^{2} + 15 a + 13\right)\cdot 17 + \left(2 a^{2} + 5 a + 5\right)\cdot 17^{2} + \left(11 a^{2} + 15 a + 6\right)\cdot 17^{3} + \left(3 a + 2\right)\cdot 17^{4} + \left(6 a^{2} + 4 a + 9\right)\cdot 17^{5} + \left(3 a^{2} + 9 a + 6\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 2 a^{2} + 9 a + 2 + \left(a^{2} + 4 a + 5\right)\cdot 17 + \left(16 a^{2} + 12 a + 1\right)\cdot 17^{2} + \left(7 a^{2} + 14 a + 12\right)\cdot 17^{3} + \left(11 a^{2} + 10 a + 6\right)\cdot 17^{4} + \left(4 a^{2} + 4 a + 3\right)\cdot 17^{5} + \left(7 a^{2} + 7 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 10 a^{2} + a + 16 + \left(16 a^{2} + 10 a\right)\cdot 17 + \left(10 a^{2} + 14 a\right)\cdot 17^{2} + \left(3 a^{2} + 10 a + 7\right)\cdot 17^{3} + \left(12 a^{2} + 12 a + 4\right)\cdot 17^{4} + \left(6 a + 11\right)\cdot 17^{5} + \left(11 a^{2} + 12 a + 11\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 16 a^{2} + 13 a + 1 + \left(2 a^{2} + a + 2\right)\cdot 17 + \left(3 a^{2} + 12 a + 13\right)\cdot 17^{2} + \left(4 a^{2} + 2\right)\cdot 17^{3} + \left(12 a^{2} + 12 a + 7\right)\cdot 17^{4} + \left(10 a^{2} + 15 a + 7\right)\cdot 17^{5} + \left(15 a^{2} + 13 a + 15\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2,8,9,5,6,4,7)$ |
| $(1,6,8)(2,7,5)(3,4,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
$c5$ |
$c6$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$3$ |
$(1,8,6)(2,5,7)(3,9,4)$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
| $1$ |
$3$ |
$(1,6,8)(2,7,5)(3,4,9)$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
$-\zeta_{9}^{3} - 1$ |
$\zeta_{9}^{3}$ |
| $1$ |
$9$ |
$(1,3,2,8,9,5,6,4,7)$ |
$\zeta_{9}$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
$\zeta_{9}^{5}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
| $1$ |
$9$ |
$(1,2,9,6,7,3,8,5,4)$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$\zeta_{9}$ |
$\zeta_{9}^{5}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ |
$9$ |
$(1,9,7,8,4,2,6,3,5)$ |
$\zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}$ |
$\zeta_{9}^{5}$ |
| $1$ |
$9$ |
$(1,5,3,6,2,4,8,7,9)$ |
$\zeta_{9}^{5}$ |
$\zeta_{9}$ |
$\zeta_{9}^{2}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
| $1$ |
$9$ |
$(1,4,5,8,3,7,6,9,2)$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$\zeta_{9}^{5}$ |
$\zeta_{9}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$\zeta_{9}^{4}$ |
$\zeta_{9}^{2}$ |
| $1$ |
$9$ |
$(1,7,4,6,5,9,8,2,3)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
$-\zeta_{9}^{4} - \zeta_{9}$ |
$\zeta_{9}^{5}$ |
$\zeta_{9}^{4}$ |
$\zeta_{9}^{2}$ |
$\zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
