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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 16\cdot 19 + 13\cdot 19^{2} + 17\cdot 19^{3} + 2\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 12 + \left(3 a + 15\right)\cdot 19 + \left(6 a + 5\right)\cdot 19^{2} + \left(17 a + 18\right)\cdot 19^{3} + \left(12 a + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 8 + \left(4 a + 18\right)\cdot 19 + \left(7 a + 6\right)\cdot 19^{2} + \left(14 a + 2\right)\cdot 19^{3} + \left(4 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 10 + \left(15 a + 2\right)\cdot 19 + \left(12 a + 8\right)\cdot 19^{2} + \left(a + 10\right)\cdot 19^{3} + \left(6 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 a + 15 + \left(14 a + 15\right)\cdot 19 + \left(11 a + 9\right)\cdot 19^{2} + \left(4 a + 9\right)\cdot 19^{3} + \left(14 a + 4\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 + 7\cdot 19 + 12\cdot 19^{2} + 17\cdot 19^{3} + 15\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,5)(3,4)$ |
| $(1,4,6,5,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $15$ | $2$ | $(2,3)(5,6)$ | $-2$ |
| $20$ | $3$ | $(1,6,2)(3,4,5)$ | $0$ |
| $30$ | $4$ | $(2,6,3,5)$ | $0$ |
| $24$ | $5$ | $(1,5,3,6,4)$ | $1$ |
| $20$ | $6$ | $(1,4,6,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
