Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(3696\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.37566144.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.231.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.231.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 6x^{4} + 9x^{2} - 11 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 14 a + 18 + \left(38 a + 22\right)\cdot 43 + \left(17 a + 12\right)\cdot 43^{2} + \left(27 a + 30\right)\cdot 43^{3} + \left(33 a + 23\right)\cdot 43^{4} + \left(22 a + 21\right)\cdot 43^{5} +O(43^{6})\)
$r_{ 2 }$ |
$=$ |
\( 36 + 16\cdot 43 + 38\cdot 43^{2} + 15\cdot 43^{3} + 32\cdot 43^{4} + 10\cdot 43^{5} +O(43^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 14 a + 11 + \left(38 a + 39\right)\cdot 43 + \left(17 a + 7\right)\cdot 43^{2} + \left(27 a + 3\right)\cdot 43^{3} + \left(33 a + 13\right)\cdot 43^{4} + \left(22 a + 32\right)\cdot 43^{5} +O(43^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 29 a + 25 + \left(4 a + 20\right)\cdot 43 + \left(25 a + 30\right)\cdot 43^{2} + \left(15 a + 12\right)\cdot 43^{3} + \left(9 a + 19\right)\cdot 43^{4} + \left(20 a + 21\right)\cdot 43^{5} +O(43^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 7 + 26\cdot 43 + 4\cdot 43^{2} + 27\cdot 43^{3} + 10\cdot 43^{4} + 32\cdot 43^{5} +O(43^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 29 a + 32 + \left(4 a + 3\right)\cdot 43 + \left(25 a + 35\right)\cdot 43^{2} + \left(15 a + 39\right)\cdot 43^{3} + \left(9 a + 29\right)\cdot 43^{4} + \left(20 a + 10\right)\cdot 43^{5} +O(43^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
$3$ | $2$ | $(1,2)(4,5)$ | $0$ |
$3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
$2$ | $3$ | $(1,6,2)(3,5,4)$ | $-1$ |
$2$ | $6$ | $(1,5,6,4,2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.