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