Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(43\) |
Artin field: | Galois closure of 6.0.147008443.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{43}(37,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 4x^{4} + 23x^{3} + 67x^{2} + 50x + 44 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: \( x^{2} + 101x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 64 a + 3 + \left(85 a + 39\right)\cdot 113 + \left(53 a + 61\right)\cdot 113^{2} + \left(66 a + 56\right)\cdot 113^{3} + \left(37 a + 13\right)\cdot 113^{4} +O(113^{5})\)
$r_{ 2 }$ |
$=$ |
\( 112 a + 89 + \left(99 a + 25\right)\cdot 113 + \left(a + 87\right)\cdot 113^{2} + \left(39 a + 52\right)\cdot 113^{3} + \left(101 a + 101\right)\cdot 113^{4} +O(113^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 74 a + 47 + \left(74 a + 24\right)\cdot 113 + \left(62 a + 6\right)\cdot 113^{2} + \left(35 a + 64\right)\cdot 113^{3} + \left(94 a + 81\right)\cdot 113^{4} +O(113^{5})\)
| $r_{ 4 }$ |
$=$ |
\( a + 77 + \left(13 a + 96\right)\cdot 113 + \left(111 a + 9\right)\cdot 113^{2} + \left(73 a + 67\right)\cdot 113^{3} + \left(11 a + 35\right)\cdot 113^{4} +O(113^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 39 a + 31 + \left(38 a + 55\right)\cdot 113 + \left(50 a + 5\right)\cdot 113^{2} + \left(77 a + 89\right)\cdot 113^{3} + \left(18 a + 47\right)\cdot 113^{4} +O(113^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 49 a + 93 + \left(27 a + 97\right)\cdot 113 + \left(59 a + 55\right)\cdot 113^{2} + \left(46 a + 9\right)\cdot 113^{3} + \left(75 a + 59\right)\cdot 113^{4} +O(113^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
$1$ | $3$ | $(1,5,4)(2,6,3)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,4,5)(2,3,6)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,2,5,6,4,3)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,3,4,6,5,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.