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:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 18 + \left(17 a^{2} + 11 a + 4\right)\cdot 19 + \left(15 a^{2} + 12 a + 8\right)\cdot 19^{2} + \left(4 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(a^{2} + 14 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 2\cdot 19 + 16\cdot 19^{2} + 17\cdot 19^{3} + 11\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a^{2} + 7 a + 11 + \left(14 a^{2} + 14 a + 8\right)\cdot 19 + \left(12 a^{2} + 16 a + 18\right)\cdot 19^{2} + \left(10 a^{2} + 9 a + 8\right)\cdot 19^{3} + \left(17 a + 9\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a^{2} + 9 a + 13 + \left(13 a^{2} + 14 a + 14\right)\cdot 19 + \left(3 a^{2} + 11 a + 13\right)\cdot 19^{2} + \left(5 a^{2} + 15 a + 14\right)\cdot 19^{3} + \left(10 a^{2} + 11 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a^{2} + 15 a + 15 + \left(3 a^{2} + 14 a + 9\right)\cdot 19 + \left(13 a^{2} + 9 a\right)\cdot 19^{2} + \left(10 a^{2} + a + 9\right)\cdot 19^{3} + \left(14 a^{2} + a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a^{2} + 13 a + 4 + \left(7 a^{2} + 11 a + 11\right)\cdot 19 + \left(18 a^{2} + 13 a + 8\right)\cdot 19^{2} + \left(8 a^{2} + 15 a + 18\right)\cdot 19^{3} + \left(7 a^{2} + 11 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 9 a^{2} + 16 a + 11 + \left(a^{2} + 8 a + 5\right)\cdot 19 + \left(12 a^{2} + 11 a + 10\right)\cdot 19^{2} + \left(16 a^{2} + 7 a + 18\right)\cdot 19^{3} + \left(3 a^{2} + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,5)(3,4)$ |
| $(1,7,2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $7$ |
| $21$ | $2$ | $(1,5)(3,4)$ | $-1$ |
| $56$ | $3$ | $(1,5,2)(3,7,4)$ | $1$ |
| $42$ | $4$ | $(1,6)(2,3,5,7)$ | $-1$ |
| $24$ | $7$ | $(1,6,5,7,2,4,3)$ | $0$ |
| $24$ | $7$ | $(1,7,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
