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