Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{5} + 5 x + 17 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a^{4} + 16 a^{3} + 8 a^{2} + 3 a + 3 + \left(5 a^{4} + 17 a^{3} + 16 a^{2} + 2 a\right)\cdot 19 + \left(17 a^{4} + 14 a^{3} + 5 a^{2} + 11 a + 14\right)\cdot 19^{2} + \left(6 a^{4} + 15 a^{3} + 6 a^{2} + 8 a + 15\right)\cdot 19^{3} + \left(4 a^{4} + a^{3} + 17 a^{2} + a + 1\right)\cdot 19^{4} + \left(10 a^{4} + 6 a^{3} + 4 a^{2} + 11 a + 7\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a^{4} + 16 a^{3} + 8 a^{2} + 3 a + 14 + \left(5 a^{4} + 17 a^{3} + 16 a^{2} + 2 a + 10\right)\cdot 19 + \left(17 a^{4} + 14 a^{3} + 5 a^{2} + 11 a + 2\right)\cdot 19^{2} + \left(6 a^{4} + 15 a^{3} + 6 a^{2} + 8 a + 9\right)\cdot 19^{3} + \left(4 a^{4} + a^{3} + 17 a^{2} + a + 6\right)\cdot 19^{4} + \left(10 a^{4} + 6 a^{3} + 4 a^{2} + 11 a + 6\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a^{4} + 5 a^{3} + 10 a^{2} + 7 + \left(15 a^{4} + 7 a^{3} + 8 a^{2} + 12 a + 13\right)\cdot 19 + \left(a^{4} + 8 a^{3} + a^{2} + 3 a + 16\right)\cdot 19^{2} + \left(6 a^{4} + 11 a^{3} + 6 a^{2} + 18 a + 5\right)\cdot 19^{3} + \left(17 a^{3} + 2 a^{2} + 5 a + 9\right)\cdot 19^{4} + \left(15 a^{4} + 16 a^{3} + 3 a^{2} + 8 a + 6\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a^{4} + 5 a^{3} + 10 a^{2} + 15 + \left(15 a^{4} + 7 a^{3} + 8 a^{2} + 12 a + 2\right)\cdot 19 + \left(a^{4} + 8 a^{3} + a^{2} + 3 a + 9\right)\cdot 19^{2} + \left(6 a^{4} + 11 a^{3} + 6 a^{2} + 18 a + 12\right)\cdot 19^{3} + \left(17 a^{3} + 2 a^{2} + 5 a + 4\right)\cdot 19^{4} + \left(15 a^{4} + 16 a^{3} + 3 a^{2} + 8 a + 7\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a^{4} + 10 a^{3} + 4 a^{2} + 18 a + 4 + \left(15 a^{4} + 8 a^{3} + 9 a^{2} + 5 a + 3\right)\cdot 19 + \left(13 a^{4} + 8 a^{3} + a^{2} + 16 a\right)\cdot 19^{2} + \left(3 a^{4} + 18 a^{3} + 14 a^{2} + 10 a + 3\right)\cdot 19^{3} + \left(4 a^{4} + 6 a^{3} + 15 a^{2} + 13 a + 1\right)\cdot 19^{4} + \left(11 a^{4} + a^{3} + a^{2} + 12 a + 11\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a^{4} + 10 a^{3} + 4 a^{2} + 18 a + 15 + \left(15 a^{4} + 8 a^{3} + 9 a^{2} + 5 a + 13\right)\cdot 19 + \left(13 a^{4} + 8 a^{3} + a^{2} + 16 a + 7\right)\cdot 19^{2} + \left(3 a^{4} + 18 a^{3} + 14 a^{2} + 10 a + 15\right)\cdot 19^{3} + \left(4 a^{4} + 6 a^{3} + 15 a^{2} + 13 a + 5\right)\cdot 19^{4} + \left(11 a^{4} + a^{3} + a^{2} + 12 a + 10\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 10 a^{4} + 13 a^{3} + 10 a^{2} + 9 a + 4 + \left(10 a^{4} + 8 a^{3} + 2 a^{2} + 4 a + 13\right)\cdot 19 + \left(17 a^{4} + a^{3} + 15 a^{2} + 2 a + 3\right)\cdot 19^{2} + \left(6 a^{4} + 18 a^{2} + 17 a + 9\right)\cdot 19^{3} + \left(14 a^{4} + 7 a^{3} + 13 a^{2} + 10 a + 8\right)\cdot 19^{4} + \left(14 a^{4} + 4 a^{3} + 5 a^{2} + 5\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 10 a^{4} + 13 a^{3} + 10 a^{2} + 9 a + 12 + \left(10 a^{4} + 8 a^{3} + 2 a^{2} + 4 a + 2\right)\cdot 19 + \left(17 a^{4} + a^{3} + 15 a^{2} + 2 a + 15\right)\cdot 19^{2} + \left(6 a^{4} + 18 a^{2} + 17 a + 15\right)\cdot 19^{3} + \left(14 a^{4} + 7 a^{3} + 13 a^{2} + 10 a + 3\right)\cdot 19^{4} + \left(14 a^{4} + 4 a^{3} + 5 a^{2} + 6\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 11 a^{4} + 13 a^{3} + 6 a^{2} + 8 a + 8 + \left(10 a^{4} + 14 a^{3} + a^{2} + 13 a + 13\right)\cdot 19 + \left(6 a^{4} + 4 a^{3} + 14 a^{2} + 4 a + 16\right)\cdot 19^{2} + \left(14 a^{4} + 11 a^{3} + 11 a^{2} + 2 a\right)\cdot 19^{3} + \left(14 a^{4} + 4 a^{3} + 7 a^{2} + 6 a + 10\right)\cdot 19^{4} + \left(5 a^{4} + 9 a^{3} + 3 a^{2} + 5 a + 7\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
| $r_{ 10 }$ |
$=$ |
$ 11 a^{4} + 13 a^{3} + 6 a^{2} + 8 a + 16 + \left(10 a^{4} + 14 a^{3} + a^{2} + 13 a + 2\right)\cdot 19 + \left(6 a^{4} + 4 a^{3} + 14 a^{2} + 4 a + 9\right)\cdot 19^{2} + \left(14 a^{4} + 11 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 19^{3} + \left(14 a^{4} + 4 a^{3} + 7 a^{2} + 6 a + 5\right)\cdot 19^{4} + \left(5 a^{4} + 9 a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 10 }$
| Cycle notation |
| $(1,3,8,9,5,2,4,7,10,6)$ |
| $(1,2)(3,4)(5,6)(7,8)(9,10)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 10 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)(9,10)$ | $-1$ |
| $1$ | $5$ | $(1,8,5,4,10)(2,7,6,3,9)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,5,10,8,4)(2,6,9,7,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,4,8,10,5)(2,3,7,9,6)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,10,4,5,8)(2,9,3,6,7)$ | $\zeta_{5}^{3}$ |
| $1$ | $10$ | $(1,3,8,9,5,2,4,7,10,6)$ | $-\zeta_{5}$ |
| $1$ | $10$ | $(1,9,4,6,8,2,10,3,5,7)$ | $-\zeta_{5}^{3}$ |
| $1$ | $10$ | $(1,7,5,3,10,2,8,6,4,9)$ | $-\zeta_{5}^{2}$ |
| $1$ | $10$ | $(1,6,10,7,4,2,5,9,8,3)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
