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