Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Artin field: | Galois closure of 6.0.2565108243.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{171}(83,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 152x^{3} + 6859 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a + 9 + \left(a + 12\right)\cdot 23 + \left(12 a + 2\right)\cdot 23^{2} + \left(14 a + 18\right)\cdot 23^{3} + \left(21 a + 1\right)\cdot 23^{4} + \left(15 a + 4\right)\cdot 23^{5} +O(23^{6})\)
| $r_{ 2 }$ |
$=$ |
\( 17 a + \left(13 a + 13\right)\cdot 23 + \left(9 a + 15\right)\cdot 23^{2} + \left(21 a + 21\right)\cdot 23^{3} + \left(7 a + 18\right)\cdot 23^{4} + \left(14 a + 11\right)\cdot 23^{5} +O(23^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 6 a + 11 + 9 a\cdot 23 + \left(13 a + 21\right)\cdot 23^{2} + \left(a + 8\right)\cdot 23^{3} + \left(15 a + 13\right)\cdot 23^{4} + \left(8 a + 9\right)\cdot 23^{5} +O(23^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 2 a + 5 + \left(21 a + 18\right)\cdot 23 + \left(10 a + 1\right)\cdot 23^{2} + \left(8 a + 12\right)\cdot 23^{3} + \left(a + 7\right)\cdot 23^{4} + \left(7 a + 14\right)\cdot 23^{5} +O(23^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 19 a + 3 + \left(11 a + 10\right)\cdot 23 + \left(20 a + 22\right)\cdot 23^{2} + \left(6 a + 18\right)\cdot 23^{3} + \left(9 a + 7\right)\cdot 23^{4} + \left(21 a + 9\right)\cdot 23^{5} +O(23^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 4 a + 18 + \left(11 a + 14\right)\cdot 23 + \left(2 a + 5\right)\cdot 23^{2} + \left(16 a + 12\right)\cdot 23^{3} + \left(13 a + 19\right)\cdot 23^{4} + \left(a + 19\right)\cdot 23^{5} +O(23^{6})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,5,4,3,6)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,6,3,4,5,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.