Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(1757\)\(\medspace = 7 \cdot 251 \) |
Artin field: | Galois closure of 6.0.37967615651.3 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{1757}(501,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 184x^{4} - 123x^{3} + 11849x^{2} - 3844x + 266111 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 2 a + 28 + 3\cdot 29 + \left(20 a + 22\right)\cdot 29^{2} + 26 a\cdot 29^{3} + \left(7 a + 19\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$ |
$=$ |
\( 2 a + 17 + 12\cdot 29 + \left(20 a + 9\right)\cdot 29^{2} + \left(26 a + 15\right)\cdot 29^{3} + \left(7 a + 24\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 2 a + 13 + \left(20 a + 7\right)\cdot 29^{2} + \left(26 a + 2\right)\cdot 29^{3} + \left(7 a + 24\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 27 a + 9 + \left(28 a + 2\right)\cdot 29 + \left(8 a + 6\right)\cdot 29^{2} + \left(2 a + 27\right)\cdot 29^{3} + \left(21 a + 2\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 27 a + 23 + \left(28 a + 27\right)\cdot 29 + \left(8 a + 19\right)\cdot 29^{2} + \left(2 a + 28\right)\cdot 29^{3} + \left(21 a + 7\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 27 a + 27 + \left(28 a + 10\right)\cdot 29 + \left(8 a + 22\right)\cdot 29^{2} + \left(2 a + 12\right)\cdot 29^{3} + \left(21 a + 8\right)\cdot 29^{4} +O(29^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
$1$ | $3$ | $(1,3,2)(4,5,6)$ | $\zeta_{3}$ |
$1$ | $3$ | $(1,2,3)(4,6,5)$ | $-\zeta_{3} - 1$ |
$1$ | $6$ | $(1,5,2,4,3,6)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,6,3,4,2,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.