Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(35\)\(\medspace = 5 \cdot 7 \) |
Artin field: | Galois closure of 6.0.2100875.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{35}(19,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 8x^{4} - 8x^{3} + 22x^{2} - 22x + 29 \) . |
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:
$r_{ 1 }$ | $=$ | \( 10 a + 19 + \left(33 a + 1\right)\cdot 41 + \left(27 a + 38\right)\cdot 41^{2} + \left(14 a + 39\right)\cdot 41^{3} + \left(9 a + 39\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 23 a + 29 + \left(40 a + 39\right)\cdot 41 + \left(22 a + 39\right)\cdot 41^{2} + \left(15 a + 2\right)\cdot 41^{3} + \left(8 a + 31\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 18 a + 16 + 15\cdot 41 + \left(18 a + 27\right)\cdot 41^{2} + \left(25 a + 26\right)\cdot 41^{3} + \left(32 a + 40\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 17 a + 21 + \left(27 a + 16\right)\cdot 41 + \left(34 a + 8\right)\cdot 41^{2} + \left(28 a + 34\right)\cdot 41^{3} + \left(7 a + 22\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 24 a + 31 + \left(13 a + 40\right)\cdot 41 + \left(6 a + 2\right)\cdot 41^{2} + \left(12 a + 4\right)\cdot 41^{3} + \left(33 a + 17\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 6 }$ | $=$ | \( 31 a + 8 + \left(7 a + 9\right)\cdot 41 + \left(13 a + 6\right)\cdot 41^{2} + \left(26 a + 15\right)\cdot 41^{3} + \left(31 a + 12\right)\cdot 41^{4} +O(41^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
$1$ | $3$ | $(1,3,4)(2,5,6)$ | $\zeta_{3}$ |
$1$ | $3$ | $(1,4,3)(2,6,5)$ | $-\zeta_{3} - 1$ |
$1$ | $6$ | $(1,5,3,6,4,2)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,2,4,6,3,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.