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