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:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 5 + \left(12 a + 16\right)\cdot 19 + \left(8 a + 2\right)\cdot 19^{2} + \left(5 a + 17\right)\cdot 19^{3} + \left(a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 9 + \left(18 a + 11\right)\cdot 19 + \left(17 a + 11\right)\cdot 19^{2} + \left(18 a + 17\right)\cdot 19^{3} + \left(7 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 4 + \left(6 a + 11\right)\cdot 19 + \left(10 a + 17\right)\cdot 19^{2} + \left(13 a + 13\right)\cdot 19^{3} + \left(17 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 6 + \left(13 a + 15\right)\cdot 19 + \left(9 a + 8\right)\cdot 19^{2} + \left(5 a + 6\right)\cdot 19^{3} + \left(12 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 17 + 2\cdot 19 + \left(a + 11\right)\cdot 19^{2} + 18\cdot 19^{3} + \left(11 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 16 + \left(5 a + 18\right)\cdot 19 + \left(9 a + 4\right)\cdot 19^{2} + \left(13 a + 2\right)\cdot 19^{3} + \left(6 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,5)(4,6)$ |
| $(1,4,5,3,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,5,3,6,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,6,3,5,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
