Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(203\)\(\medspace = 7 \cdot 29 \) |
Artin field: | Galois closure of 6.6.58557989.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | even |
Dirichlet character: | \(\chi_{203}(86,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 26x^{4} + 17x^{3} + 159x^{2} - 64x - 169 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 38 a + 19 + \left(37 a + 33\right)\cdot 41 + \left(19 a + 26\right)\cdot 41^{2} + \left(23 a + 1\right)\cdot 41^{3} + \left(26 a + 22\right)\cdot 41^{4} +O(41^{5})\)
$r_{ 2 }$ |
$=$ |
\( 3 a + 33 + \left(3 a + 23\right)\cdot 41 + \left(21 a + 25\right)\cdot 41^{2} + \left(17 a + 36\right)\cdot 41^{3} + \left(14 a + 17\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 38 a + 35 + \left(37 a + 27\right)\cdot 41 + \left(19 a + 18\right)\cdot 41^{2} + \left(23 a + 18\right)\cdot 41^{3} + \left(26 a + 34\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 3 a + 26 + \left(3 a + 21\right)\cdot 41 + \left(21 a + 40\right)\cdot 41^{2} + \left(17 a + 27\right)\cdot 41^{3} + \left(14 a + 8\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 3 a + 10 + \left(3 a + 27\right)\cdot 41 + \left(21 a + 7\right)\cdot 41^{2} + \left(17 a + 11\right)\cdot 41^{3} + \left(14 a + 37\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 38 a + 1 + \left(37 a + 30\right)\cdot 41 + \left(19 a + 3\right)\cdot 41^{2} + \left(23 a + 27\right)\cdot 41^{3} + \left(26 a + 2\right)\cdot 41^{4} +O(41^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
$1$ | $3$ | $(1,3,6)(2,5,4)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,6,3)(2,4,5)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,2,3,5,6,4)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,4,6,5,3,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.