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