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