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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 14 + \left(9 a + 10\right)\cdot 19 + \left(5 a + 15\right)\cdot 19^{2} + \left(14 a + 6\right)\cdot 19^{3} + \left(6 a + 13\right)\cdot 19^{4} + \left(11 a + 7\right)\cdot 19^{5} + \left(4 a + 12\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + 12 + \left(a + 2\right)\cdot 19 + \left(6 a + 8\right)\cdot 19^{2} + \left(12 a + 15\right)\cdot 19^{3} + \left(8 a + 7\right)\cdot 19^{4} + \left(4 a + 18\right)\cdot 19^{5} + 9\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 3\cdot 19 + 18\cdot 19^{2} + 9\cdot 19^{3} + 5\cdot 19^{4} + 7\cdot 19^{5} + 10\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 1 + \left(17 a + 9\right)\cdot 19 + 18\cdot 19^{2} + \left(5 a + 3\right)\cdot 19^{3} + \left(7 a + 17\right)\cdot 19^{4} + 11 a\cdot 19^{5} + \left(4 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 9 + \left(17 a + 7\right)\cdot 19 + \left(12 a + 12\right)\cdot 19^{2} + \left(6 a + 2\right)\cdot 19^{3} + \left(10 a + 4\right)\cdot 19^{4} + \left(14 a + 14\right)\cdot 19^{5} + \left(18 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 + 5\cdot 19 + 9\cdot 19^{2} + 13\cdot 19^{3} + 2\cdot 19^{4} + 10\cdot 19^{5} + 9\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 5 a + 15 + \left(a + 12\right)\cdot 19 + \left(18 a + 1\right)\cdot 19^{2} + \left(13 a + 8\right)\cdot 19^{3} + 11 a\cdot 19^{4} + \left(7 a + 5\right)\cdot 19^{5} + \left(14 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 4 a + 10 + \left(9 a + 5\right)\cdot 19 + \left(13 a + 11\right)\cdot 19^{2} + \left(4 a + 15\right)\cdot 19^{3} + \left(12 a + 5\right)\cdot 19^{4} + \left(7 a + 12\right)\cdot 19^{5} + \left(14 a + 5\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(2,6,5)(3,4,7)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(2,3)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $12$ | $2$ | $(2,3)(4,5)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,3,2)(6,7,8)$ | $1$ |
| $6$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $8$ | $6$ | $(1,8)(2,3,5,7,6,4)$ | $-1$ |
| $6$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
| $6$ | $8$ | $(1,3,5,2,8,6,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
