Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
Artin field: | Galois closure of 6.6.190102016.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | even |
Dirichlet character: | \(\chi_{104}(69,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 26x^{4} + 104x^{2} - 104 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 22 a + 9 + \left(18 a + 23\right)\cdot 31 + \left(28 a + 11\right)\cdot 31^{2} + 13 a\cdot 31^{3} + \left(8 a + 14\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 12 a + 19 + \left(7 a + 29\right)\cdot 31 + \left(6 a + 12\right)\cdot 31^{2} + \left(6 a + 12\right)\cdot 31^{3} + \left(24 a + 25\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 14 a + 17 + \left(22 a + 15\right)\cdot 31 + \left(23 a + 18\right)\cdot 31^{2} + \left(3 a + 23\right)\cdot 31^{3} + \left(11 a + 21\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 9 a + 22 + \left(12 a + 7\right)\cdot 31 + \left(2 a + 19\right)\cdot 31^{2} + \left(17 a + 30\right)\cdot 31^{3} + \left(22 a + 16\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 19 a + 12 + \left(23 a + 1\right)\cdot 31 + \left(24 a + 18\right)\cdot 31^{2} + \left(24 a + 18\right)\cdot 31^{3} + \left(6 a + 5\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 17 a + 14 + \left(8 a + 15\right)\cdot 31 + \left(7 a + 12\right)\cdot 31^{2} + \left(27 a + 7\right)\cdot 31^{3} + \left(19 a + 9\right)\cdot 31^{4} +O(31^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
$1$ | $3$ | $(1,3,5)(2,4,6)$ | $\zeta_{3}$ |
$1$ | $3$ | $(1,5,3)(2,6,4)$ | $-\zeta_{3} - 1$ |
$1$ | $6$ | $(1,2,3,4,5,6)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,6,5,4,3,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.