Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 5 + \left(9 a + 16\right)\cdot 19 + \left(18 a + 5\right)\cdot 19^{2} + \left(8 a + 8\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} + \left(11 a + 3\right)\cdot 19^{5} + \left(10 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 19 + 4\cdot 19^{2} + 14\cdot 19^{3} + 5\cdot 19^{4} + 11\cdot 19^{5} + 18\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 9 + \left(12 a + 10\right)\cdot 19 + \left(6 a + 9\right)\cdot 19^{2} + \left(18 a + 2\right)\cdot 19^{3} + \left(7 a + 11\right)\cdot 19^{4} + \left(17 a + 2\right)\cdot 19^{5} + \left(4 a + 11\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 7 + \left(6 a + 6\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + 14\cdot 19^{3} + 11 a\cdot 19^{4} + \left(a + 12\right)\cdot 19^{5} + \left(14 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 8 + \left(9 a + 3\right)\cdot 19 + 15\cdot 19^{2} + \left(10 a + 17\right)\cdot 19^{3} + \left(13 a + 17\right)\cdot 19^{4} + \left(7 a + 8\right)\cdot 19^{5} + \left(8 a + 4\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
