Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.137200.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.35.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.140.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{4} - x^{3} + 2x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 17 + \left(8 a + 8\right)\cdot 23 + \left(16 a + 4\right)\cdot 23^{2} + \left(a + 18\right)\cdot 23^{3} + \left(15 a + 14\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 2 a + 8 + \left(18 a + 20\right)\cdot 23 + \left(2 a + 14\right)\cdot 23^{2} + \left(4 a + 1\right)\cdot 23^{3} + \left(18 a + 15\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 13 + 18\cdot 23 + 17\cdot 23^{2} + 11\cdot 23^{3} +O(23^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 8 a + 1 + \left(14 a + 11\right)\cdot 23 + \left(6 a + 5\right)\cdot 23^{2} + \left(21 a + 5\right)\cdot 23^{3} + \left(7 a + 20\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 19 + 23 + 23^{2} + 2\cdot 23^{3} + 17\cdot 23^{4} +O(23^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 21 a + 12 + \left(4 a + 8\right)\cdot 23 + \left(20 a + 2\right)\cdot 23^{2} + \left(18 a + 7\right)\cdot 23^{3} + \left(4 a + 1\right)\cdot 23^{4} +O(23^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,4,5)(2,3,6)$ | $-1$ |
| $2$ | $6$ | $(1,3,4,6,5,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.