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