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:
| $r_{ 1 }$ |
$=$ |
$ 17 a^{2} + 18 a + 4 + \left(18 a^{2} + 5 a + 15\right)\cdot 19 + \left(9 a^{2} + 6 a + 7\right)\cdot 19^{2} + \left(2 a^{2} + 15 a + 18\right)\cdot 19^{3} + \left(12 a^{2} + 18 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a^{2} + 14 a + 16 + \left(4 a^{2} + 16 a + 8\right)\cdot 19 + \left(8 a^{2} + 3 a + 9\right)\cdot 19^{2} + \left(a^{2} + 9 a + 15\right)\cdot 19^{3} + \left(a^{2} + 16 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 6\cdot 19 + 9\cdot 19^{2} + 16\cdot 19^{3} + 9\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a^{2} + 17 a + 9 + \left(5 a^{2} + 17 a + 4\right)\cdot 19 + \left(4 a^{2} + 14 a + 8\right)\cdot 19^{2} + \left(5 a^{2} + 15 a + 9\right)\cdot 19^{3} + \left(a^{2} + 5 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a^{2} + 6 a + 8 + \left(14 a^{2} + 15 a + 3\right)\cdot 19 + \left(8 a + 2\right)\cdot 19^{2} + \left(15 a^{2} + 13 a + 1\right)\cdot 19^{3} + \left(5 a^{2} + 2 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a^{2} + 17 a + 13 + \left(8 a^{2} + 17 a + 17\right)\cdot 19 + \left(2 a^{2} + 13 a + 15\right)\cdot 19^{2} + \left(7 a^{2} + 2 a + 1\right)\cdot 19^{3} + \left(15 a^{2} + 16 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 12 a^{2} + 4 a + 2 + \left(4 a^{2} + 2 a + 1\right)\cdot 19 + \left(12 a^{2} + 9 a + 4\right)\cdot 19^{2} + \left(6 a^{2} + 13\right)\cdot 19^{3} + \left(2 a^{2} + 16 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,5)(2,3)$ |
| $(1,6,4,2)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,5)(2,3)$ |
$2$ |
| $56$ |
$3$ |
$(1,4,3)(2,7,6)$ |
$0$ |
| $42$ |
$4$ |
$(1,6,4,2)(5,7)$ |
$0$ |
| $24$ |
$7$ |
$(1,7,5,6,4,2,3)$ |
$-1$ |
| $24$ |
$7$ |
$(1,6,3,5,2,7,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
