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