Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
Artin field: | Galois closure of 6.0.43243551.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{117}(38,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} + 18x^{4} - 10x^{3} + 81x^{2} - 90x + 181 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 12 a + 4 + \left(10 a + 15\right)\cdot 17 + \left(16 a + 12\right)\cdot 17^{2} + \left(10 a + 1\right)\cdot 17^{3} + \left(11 a + 2\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 15 a + 6 + \left(10 a + 8\right)\cdot 17 + \left(10 a + 11\right)\cdot 17^{2} + \left(10 a + 12\right)\cdot 17^{3} + \left(14 a + 4\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 2 a + 4 + \left(6 a + 4\right)\cdot 17 + \left(6 a + 11\right)\cdot 17^{2} + \left(6 a + 12\right)\cdot 17^{3} + \left(2 a + 8\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 4 }$ | $=$ | \( 3 a + 9 + 14\cdot 17 + \left(11 a + 9\right)\cdot 17^{2} + \left(16 a + 2\right)\cdot 17^{3} + \left(2 a + 6\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 5 }$ | $=$ | \( 14 a + 12 + \left(16 a + 11\right)\cdot 17 + \left(5 a + 3\right)\cdot 17^{2} + 8\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 6 }$ | $=$ | \( 5 a + 16 + \left(6 a + 13\right)\cdot 17 + 17^{2} + \left(6 a + 13\right)\cdot 17^{3} + \left(5 a + 2\right)\cdot 17^{4} +O(17^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
$1$ | $3$ | $(1,3,4)(2,5,6)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,4,3)(2,6,5)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,5,3,6,4,2)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,2,4,6,3,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.