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