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