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