Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3332\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.77715568.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.68.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.3332.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 5x^{4} + 15x^{3} - 10x^{2} + 2x + 92 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 + 15\cdot 37 + 22\cdot 37^{2} + 21\cdot 37^{3} + 20\cdot 37^{4} + 36\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 23 + \left(15 a + 7\right)\cdot 37 + \left(9 a + 23\right)\cdot 37^{2} + \left(32 a + 2\right)\cdot 37^{3} + \left(6 a + 35\right)\cdot 37^{4} + \left(18 a + 19\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + \left(15 a + 18\right)\cdot 37 + \left(9 a + 28\right)\cdot 37^{2} + \left(32 a + 25\right)\cdot 37^{3} + \left(6 a + 6\right)\cdot 37^{4} + \left(18 a + 25\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 15 + \left(21 a + 29\right)\cdot 37 + \left(27 a + 13\right)\cdot 37^{2} + \left(4 a + 34\right)\cdot 37^{3} + \left(30 a + 1\right)\cdot 37^{4} + \left(18 a + 17\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 + 21\cdot 37 + 14\cdot 37^{2} + 15\cdot 37^{3} + 16\cdot 37^{4} +O(37^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 1 + \left(21 a + 19\right)\cdot 37 + \left(27 a + 8\right)\cdot 37^{2} + \left(4 a + 11\right)\cdot 37^{3} + \left(30 a + 30\right)\cdot 37^{4} + \left(18 a + 11\right)\cdot 37^{5} +O(37^{6})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ | |
| $3$ | $2$ | $(2,6)(3,4)$ | $0$ | |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,3,4)(2,5,6)$ | $-1$ | |
| $2$ | $6$ | $(1,2,3,5,4,6)$ | $1$ |