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