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