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^{3} + x + 14 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a^{2} + 14 a + 1 + \left(5 a^{2} + 6 a + 10\right)\cdot 17 + \left(2 a^{2} + 10 a + 1\right)\cdot 17^{2} + \left(3 a^{2} + 11 a + 7\right)\cdot 17^{3} + \left(6 a^{2} + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a^{2} + 6 a + 3 + \left(5 a^{2} + 14 a + 10\right)\cdot 17 + \left(a^{2} + 14 a + 6\right)\cdot 17^{2} + \left(15 a^{2} + 2 a + 9\right)\cdot 17^{3} + \left(6 a^{2} + a\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 13\cdot 17 + 13\cdot 17^{2} + 3\cdot 17^{3} + 11\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a^{2} + 6 a + 8 + \left(4 a^{2} + 10 a + 3\right)\cdot 17 + \left(a^{2} + 13\right)\cdot 17^{2} + \left(15 a^{2} + 16 a + 3\right)\cdot 17^{3} + \left(7 a^{2} + 16 a + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a^{2} + 15 a + 15 + \left(2 a^{2} + 11 a + 12\right)\cdot 17 + \left(4 a^{2} + 3\right)\cdot 17^{2} + \left(8 a^{2} + a + 16\right)\cdot 17^{3} + \left(2 a + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a^{2} + 14 a + 14 + \left(6 a^{2} + 12 a + 16\right)\cdot 17 + \left(13 a^{2} + 8 a + 8\right)\cdot 17^{2} + \left(15 a^{2} + 2 a + 15\right)\cdot 17^{3} + \left(3 a^{2} + 15 a + 9\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a + 3 + \left(10 a^{2} + 11 a + 1\right)\cdot 17 + \left(11 a^{2} + 15 a + 3\right)\cdot 17^{2} + \left(10 a^{2} + 16 a + 12\right)\cdot 17^{3} + \left(8 a^{2} + 14 a + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,3,7,5)(4,6)$ |
| $(1,6)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,6)(2,3)$ | $-1$ |
| $56$ | $3$ | $(1,7,2)(3,4,5)$ | $0$ |
| $42$ | $4$ | $(1,3,7,5)(4,6)$ | $1$ |
| $24$ | $7$ | $(1,4,6,3,2,7,5)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ | $7$ | $(1,3,5,6,7,4,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.
