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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 16 + \left(14 a + 17\right)\cdot 19 + \left(13 a + 11\right)\cdot 19^{3} + \left(16 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 14 + \left(4 a + 11\right)\cdot 19 + \left(18 a + 16\right)\cdot 19^{2} + \left(5 a + 14\right)\cdot 19^{3} + \left(2 a + 9\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 1 + \left(4 a + 5\right)\cdot 19 + \left(18 a + 4\right)\cdot 19^{2} + \left(5 a + 18\right)\cdot 19^{3} + \left(2 a + 9\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 18 + \left(4 a + 10\right)\cdot 19 + \left(18 a + 6\right)\cdot 19^{2} + \left(5 a + 4\right)\cdot 19^{3} + \left(2 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 12 + \left(14 a + 18\right)\cdot 19 + 10\cdot 19^{2} + \left(13 a + 2\right)\cdot 19^{3} + \left(16 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 18 + \left(14 a + 11\right)\cdot 19 + 17\cdot 19^{2} + \left(13 a + 5\right)\cdot 19^{3} + \left(16 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,3,5,4,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
