Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{3} + 2 x + 18 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a^{2} + a + 1 + \left(9 a^{2} + 20 a + 9\right)\cdot 23 + \left(20 a^{2} + a + 9\right)\cdot 23^{2} + \left(2 a^{2} + 9 a + 4\right)\cdot 23^{3} + \left(2 a^{2} + 10 a + 21\right)\cdot 23^{4} + \left(13 a^{2} + 7 a + 22\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a^{2} + 6 a + 2 + \left(13 a^{2} + 19 a + 14\right)\cdot 23 + \left(12 a^{2} + 4 a + 19\right)\cdot 23^{2} + \left(4 a^{2} + 14 a + 1\right)\cdot 23^{3} + \left(21 a^{2} + 4 a + 5\right)\cdot 23^{4} + \left(10 a^{2} + 3 a + 18\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a^{2} + 21 a + 17 + \left(22 a^{2} + 15 a + 2\right)\cdot 23 + \left(4 a^{2} + 7 a + 17\right)\cdot 23^{2} + \left(10 a^{2} + 20 a + 1\right)\cdot 23^{3} + \left(15 a^{2} + 22 a + 5\right)\cdot 23^{4} + \left(5 a^{2} + 18 a + 11\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a^{2} + 10 a + 11 + \left(2 a^{2} + 7 a + 22\right)\cdot 23 + \left(10 a^{2} + 16 a + 10\right)\cdot 23^{2} + \left(8 a^{2} + 15 a + 19\right)\cdot 23^{3} + \left(14 a^{2} + 17 a + 6\right)\cdot 23^{4} + \left(21 a^{2} + 11\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 22\cdot 23 + 21\cdot 23^{2} + 10\cdot 23^{3} + 14\cdot 23^{4} + 18\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a^{2} + 19 a + 19 + \left(9 a^{2} + 10 a + 16\right)\cdot 23 + \left(5 a^{2} + 10 a + 17\right)\cdot 23^{2} + \left(8 a^{2} + 11 a + 6\right)\cdot 23^{3} + \left(9 a^{2} + 18 a + 12\right)\cdot 23^{4} + \left(6 a^{2} + 4\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 15 a^{2} + 12 a + 13 + \left(11 a^{2} + 18 a + 4\right)\cdot 23 + \left(15 a^{2} + 4 a + 18\right)\cdot 23^{2} + \left(11 a^{2} + 21 a\right)\cdot 23^{3} + \left(6 a^{2} + 17 a + 4\right)\cdot 23^{4} + \left(11 a^{2} + 14 a + 5\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,6,7,2)(4,5)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $56$ | $3$ | $(1,7,3)(2,5,6)$ | $0$ |
| $42$ | $4$ | $(1,6,7,2)(4,5)$ | $1$ |
| $24$ | $7$ | $(1,5,4,6,7,2,3)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ | $7$ | $(1,6,3,4,2,5,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.
