Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(82979\)\(\medspace = 13^{2} \cdot 491 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.529654957.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.491.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.491.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 8x^{4} - 3x^{3} + 35x^{2} - 100x - 16 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 19\cdot 23 + 15\cdot 23^{2} + 13\cdot 23^{3} + 10\cdot 23^{4} + 10\cdot 23^{5} + 9\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 9 + \left(6 a + 22\right)\cdot 23 + \left(21 a + 12\right)\cdot 23^{2} + \left(3 a + 6\right)\cdot 23^{3} + \left(20 a + 5\right)\cdot 23^{4} + \left(12 a + 21\right)\cdot 23^{5} + \left(12 a + 2\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 3 + \left(6 a + 1\right)\cdot 23 + \left(21 a + 19\right)\cdot 23^{2} + \left(3 a + 2\right)\cdot 23^{3} + \left(20 a + 5\right)\cdot 23^{4} + \left(12 a + 19\right)\cdot 23^{5} + \left(12 a + 16\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 22 + \left(16 a + 16\right)\cdot 23 + \left(a + 8\right)\cdot 23^{2} + \left(19 a + 12\right)\cdot 23^{3} + \left(2 a + 18\right)\cdot 23^{4} + \left(10 a + 1\right)\cdot 23^{5} + \left(10 a + 6\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 5 + \left(16 a + 15\right)\cdot 23 + \left(a + 2\right)\cdot 23^{2} + \left(19 a + 16\right)\cdot 23^{3} + \left(2 a + 18\right)\cdot 23^{4} + \left(10 a + 3\right)\cdot 23^{5} + \left(10 a + 15\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 17\cdot 23 + 9\cdot 23^{2} + 17\cdot 23^{3} + 10\cdot 23^{4} + 12\cdot 23^{5} + 18\cdot 23^{6} +O(23^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-2$ |
| $3$ | $2$ | $(2,5)(3,4)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,2,4,6,3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.