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} + 17 a + 9 + \left(4 a^{2} + 15 a + 7\right)\cdot 19 + \left(18 a^{2} + 15 a + 17\right)\cdot 19^{2} + \left(12 a^{2} + 16 a + 11\right)\cdot 19^{3} + \left(6 a^{2} + 6 a + 18\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a^{2} + 16 a + 15 + \left(2 a^{2} + 3 a + 2\right)\cdot 19 + \left(17 a^{2} + 7 a + 14\right)\cdot 19^{2} + \left(4 a^{2} + 11 a + 17\right)\cdot 19^{3} + \left(9 a^{2} + 14 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a^{2} + 16 a + 2 + \left(17 a^{2} + 14 a + 3\right)\cdot 19 + \left(8 a + 15\right)\cdot 19^{2} + \left(2 a^{2} + 15 a + 3\right)\cdot 19^{3} + \left(16 a^{2} + 8 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 + 12\cdot 19 + 17\cdot 19^{2} + 15\cdot 19^{3} + 2\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a^{2} + 17 a + 13 + 14\cdot 19 + \left(8 a^{2} + 9 a + 2\right)\cdot 19^{2} + \left(10 a^{2} + a + 5\right)\cdot 19^{3} + \left(6 a^{2} + 16 a + 18\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 a^{2} + 6 a + 14 + \left(17 a^{2} + 17\right)\cdot 19 + \left(3 a + 8\right)\cdot 19^{2} + \left(12 a^{2} + 11 a + 11\right)\cdot 19^{3} + \left(12 a^{2} + 14 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 12 a^{2} + 4 a + 2 + \left(13 a^{2} + 2 a + 18\right)\cdot 19 + \left(11 a^{2} + 13 a + 18\right)\cdot 19^{2} + \left(14 a^{2} + 9\right)\cdot 19^{3} + \left(5 a^{2} + 15 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,5,6,3)(2,4)$ |
| $(2,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $21$ |
$2$ |
$(2,5)(6,7)$ |
$-1$ |
$-1$ |
| $56$ |
$3$ |
$(1,6,7)(2,3,5)$ |
$0$ |
$0$ |
| $42$ |
$4$ |
$(1,5,6,3)(2,4)$ |
$1$ |
$1$ |
| $24$ |
$7$ |
$(1,5,4,2,6,7,3)$ |
$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ |
$7$ |
$(1,2,3,4,7,5,6)$ |
$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.
