Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
Artin field: | Galois closure of 6.6.168488679177.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | even |
Dirichlet character: | \(\chi_{273}(17,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 83x^{4} + 433x^{3} - 566x^{2} - 120x + 64 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 8 a + 3 + \left(7 a + 9\right)\cdot 23 + \left(11 a + 22\right)\cdot 23^{2} + \left(9 a + 18\right)\cdot 23^{3} + \left(9 a + 9\right)\cdot 23^{4} +O(23^{5})\)
$r_{ 2 }$ |
$=$ |
\( 15 a + 19 + \left(15 a + 15\right)\cdot 23 + \left(11 a + 14\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(13 a + 19\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 10 a + 8 + \left(9 a + 6\right)\cdot 23 + \left(22 a + 14\right)\cdot 23^{2} + \left(10 a + 8\right)\cdot 23^{3} + \left(7 a + 16\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 13 a + 5 + \left(13 a + 15\right)\cdot 23 + 3\cdot 23^{2} + \left(12 a + 8\right)\cdot 23^{3} + \left(15 a + 20\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 6 a + \left(13 a + 1\right)\cdot 23 + \left(17 a + 19\right)\cdot 23^{2} + \left(20 a + 2\right)\cdot 23^{3} + \left(16 a + 18\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 17 a + 12 + \left(9 a + 21\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(2 a + 3\right)\cdot 23^{3} + \left(6 a + 8\right)\cdot 23^{4} +O(23^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$1$ | $3$ | $(1,5,4)(2,6,3)$ | $\zeta_{3}$ |
$1$ | $3$ | $(1,4,5)(2,3,6)$ | $-\zeta_{3} - 1$ |
$1$ | $6$ | $(1,3,5,2,4,6)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,6,4,2,5,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.