Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 9 + \left(13 a + 14\right)\cdot 17 + 5\cdot 17^{2} + \left(5 a + 13\right)\cdot 17^{3} + \left(a + 13\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 10 + 14 a\cdot 17 + 15 a\cdot 17^{2} + \left(12 a + 16\right)\cdot 17^{3} + \left(a + 13\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 3\cdot 17 + 4\cdot 17^{2} + 7\cdot 17^{3} + 2\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 8 + 2 a\cdot 17 + \left(a + 1\right)\cdot 17^{2} + \left(4 a + 13\right)\cdot 17^{3} + \left(15 a + 2\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 14 + \left(11 a + 8\right)\cdot 17 + 3\cdot 17^{2} + \left(11 a + 12\right)\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 7 + \left(3 a + 13\right)\cdot 17 + \left(16 a + 9\right)\cdot 17^{2} + 11 a\cdot 17^{3} + \left(15 a + 10\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 a + 8 + \left(5 a + 9\right)\cdot 17 + \left(16 a + 9\right)\cdot 17^{2} + \left(5 a + 5\right)\cdot 17^{3} + \left(4 a + 13\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,5)(3,6)(4,7)$ |
| $(1,3)(2,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $7$ |
$2$ |
$(1,5)(3,6)(4,7)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$7$ |
$(1,2,5,3,7,4,6)$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ |
$7$ |
$(1,5,7,6,2,3,4)$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ |
$7$ |
$(1,3,6,5,4,2,7)$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
