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^{3} + 4 x + 17 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a^{2} + 10 a + 18 + \left(17 a^{2} + 15 a + 9\right)\cdot 19 + \left(12 a^{2} + 15 a + 11\right)\cdot 19^{2} + \left(11 a^{2} + 14 a + 7\right)\cdot 19^{3} + \left(2 a^{2} + 7 a + 3\right)\cdot 19^{4} + \left(16 a + 18\right)\cdot 19^{5} + \left(10 a^{2} + 3 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 12\cdot 19 + 11\cdot 19^{2} + 4\cdot 19^{3} + 19^{4} + 4\cdot 19^{5} + 11\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a^{2} + 12 a + 12 + \left(a^{2} + 13 a\right)\cdot 19 + \left(12 a^{2} + 12 a + 3\right)\cdot 19^{2} + \left(12 a^{2} + 15 a + 10\right)\cdot 19^{3} + \left(9 a + 4\right)\cdot 19^{4} + \left(10 a^{2} + 18 a\right)\cdot 19^{5} + \left(8 a^{2} + 3 a + 14\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a^{2} + 16 a + 16 + \left(8 a + 15\right)\cdot 19 + \left(13 a^{2} + 9 a + 11\right)\cdot 19^{2} + \left(13 a^{2} + 7 a + 6\right)\cdot 19^{3} + \left(15 a^{2} + a\right)\cdot 19^{4} + \left(8 a^{2} + 3 a + 16\right)\cdot 19^{5} + \left(11 a + 17\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a^{2} + a + 2 + \left(7 a^{2} + 9 a + 15\right)\cdot 19 + \left(16 a^{2} + 2 a + 7\right)\cdot 19^{2} + \left(18 a^{2} + 5 a + 7\right)\cdot 19^{3} + \left(15 a + 5\right)\cdot 19^{4} + \left(4 a^{2} + a + 6\right)\cdot 19^{5} + \left(6 a^{2} + 6 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a^{2} + 14 a + 7 + \left(a^{2} + 4 a + 10\right)\cdot 19 + \left(a^{2} + 4 a + 17\right)\cdot 19^{2} + \left(5 a^{2} + 6 a + 14\right)\cdot 19^{3} + \left(16 a^{2} + 8 a + 1\right)\cdot 19^{4} + \left(17 a^{2} + 11 a + 5\right)\cdot 19^{5} + \left(10 a^{2} + 6 a + 2\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 15 + 16\cdot 19 + 12\cdot 19^{2} + 4\cdot 19^{3} + 19^{4} + 13\cdot 19^{5} +O\left(19^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 16 a^{2} + 4 a + 2 + \left(9 a^{2} + 5 a + 14\right)\cdot 19 + \left(a^{2} + 12 a + 18\right)\cdot 19^{2} + \left(14 a^{2} + 7 a\right)\cdot 19^{3} + \left(a^{2} + 14 a + 1\right)\cdot 19^{4} + \left(16 a^{2} + 5 a + 13\right)\cdot 19^{5} + \left(a^{2} + 6 a + 9\right)\cdot 19^{6} +O\left(19^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,6,8,3)(2,5,7,4)$ |
| $(2,5,3)(4,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $4$ |
$3$ |
$(1,5,7)(2,8,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $4$ |
$3$ |
$(1,7,5)(2,4,8)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $6$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
$0$ |
| $4$ |
$6$ |
$(1,6,4,8,3,5)(2,7)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $4$ |
$6$ |
$(1,5,3,8,4,6)(2,7)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
