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:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 9 + \left(12 a + 7\right)\cdot 17 + \left(10 a + 11\right)\cdot 17^{2} + \left(7 a + 9\right)\cdot 17^{3} + \left(12 a + 6\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 11\cdot 17 + 5\cdot 17^{2} + 10\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 8 + \left(a + 13\right)\cdot 17 + \left(9 a + 4\right)\cdot 17^{2} + \left(a + 15\right)\cdot 17^{3} + a\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 8 + \left(4 a + 4\right)\cdot 17 + \left(6 a + 9\right)\cdot 17^{2} + \left(9 a + 6\right)\cdot 17^{3} + \left(4 a + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 + 6\cdot 17 + 9\cdot 17^{2} + 2\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 + 4\cdot 17 + 15\cdot 17^{2} + 15\cdot 17^{3} + 11\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 5 a + 3 + \left(15 a + 3\right)\cdot 17 + \left(7 a + 12\right)\cdot 17^{2} + \left(15 a + 7\right)\cdot 17^{3} + 15 a\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
