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