Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a + 8 + \left(5 a + 8\right)\cdot 19 + \left(12 a + 6\right)\cdot 19^{2} + \left(3 a + 15\right)\cdot 19^{3} + \left(12 a + 14\right)\cdot 19^{4} + \left(16 a + 12\right)\cdot 19^{5} + \left(11 a + 5\right)\cdot 19^{6} + \left(18 a + 18\right)\cdot 19^{7} + \left(3 a + 11\right)\cdot 19^{8} + \left(8 a + 11\right)\cdot 19^{9} + \left(14 a + 8\right)\cdot 19^{10} +O\left(19^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 10\cdot 19 + 9\cdot 19^{2} + 7\cdot 19^{3} + 18\cdot 19^{4} + 19^{5} + 11\cdot 19^{6} + 15\cdot 19^{7} + 11\cdot 19^{8} + 5\cdot 19^{9} + 14\cdot 19^{10} +O\left(19^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 8 + \left(13 a + 4\right)\cdot 19 + \left(8 a + 11\right)\cdot 19^{2} + \left(9 a + 7\right)\cdot 19^{3} + \left(9 a + 9\right)\cdot 19^{4} + \left(2 a + 7\right)\cdot 19^{5} + \left(9 a + 12\right)\cdot 19^{6} + \left(18 a + 11\right)\cdot 19^{7} + \left(15 a + 15\right)\cdot 19^{8} + \left(3 a + 1\right)\cdot 19^{9} + \left(14 a + 2\right)\cdot 19^{10} +O\left(19^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 9 + \left(13 a + 12\right)\cdot 19 + \left(6 a + 13\right)\cdot 19^{2} + \left(15 a + 6\right)\cdot 19^{3} + \left(6 a + 4\right)\cdot 19^{4} + \left(2 a + 17\right)\cdot 19^{5} + 7 a\cdot 19^{6} + 6\cdot 19^{7} + \left(15 a + 16\right)\cdot 19^{8} + \left(10 a + 15\right)\cdot 19^{9} + \left(4 a + 14\right)\cdot 19^{10} +O\left(19^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 19 + 14\cdot 19^{2} + 5\cdot 19^{3} + 17\cdot 19^{4} + 13\cdot 19^{5} + 14\cdot 19^{6} + 10\cdot 19^{7} + 11\cdot 19^{8} + 17\cdot 19^{9} + 19^{10} +O\left(19^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 13 + \left(5 a + 12\right)\cdot 19 + \left(10 a + 6\right)\cdot 19^{2} + \left(9 a + 8\right)\cdot 19^{3} + \left(9 a + 9\right)\cdot 19^{4} + 16 a\cdot 19^{5} + 9 a\cdot 19^{6} + 2\cdot 19^{7} + \left(3 a + 13\right)\cdot 19^{8} + \left(15 a + 8\right)\cdot 19^{9} + \left(4 a + 12\right)\cdot 19^{10} +O\left(19^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 11 + 10\cdot 19 + 10\cdot 19^{2} + 4\cdot 19^{3} + 14\cdot 19^{4} + 6\cdot 19^{5} + 2\cdot 19^{6} + 13\cdot 19^{7} + 3\cdot 19^{8} + 8\cdot 19^{9} + 16\cdot 19^{10} +O\left(19^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 3 + 15\cdot 19 + 3\cdot 19^{2} + 19^{3} + 7\cdot 19^{4} + 15\cdot 19^{5} + 9\cdot 19^{6} + 17\cdot 19^{7} + 10\cdot 19^{8} + 6\cdot 19^{9} + 5\cdot 19^{10} +O\left(19^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(3,6)$ |
| $(1,6,4,3)(2,5)$ |
| $(1,7)(2,6)(3,5)(4,8)$ |
| $(1,4)(2,5)(3,6)(7,8)$ |
| $(1,6)(2,7)(3,4)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)(7,8)$ |
$-4$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,4)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,8)(3,4)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(3,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,6)(3,5)(4,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,6,5)(2,4,8,3)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,6,7)(2,3,8,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,4,3)(2,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,4,6)(2,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,4,2)(3,7,6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
