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