Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(16400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.65600.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.41.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.65600.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 2x^{3} + 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 a + 29 + \left(16 a + 5\right)\cdot 37 + \left(9 a + 36\right)\cdot 37^{2} + \left(7 a + 2\right)\cdot 37^{3} + \left(10 a + 36\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 34 + \left(17 a + 24\right)\cdot 37 + \left(13 a + 35\right)\cdot 37^{2} + 17 a\cdot 37^{3} + \left(18 a + 7\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(27 a + 20\right)\cdot 37^{2} + \left(29 a + 22\right)\cdot 37^{3} + \left(26 a + 32\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 13\cdot 37 + 3\cdot 37^{2} + 16\cdot 37^{3} + 3\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 15\cdot 37 + 17\cdot 37^{2} + 11\cdot 37^{3} + 5\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 15 + \left(19 a + 35\right)\cdot 37 + \left(23 a + 34\right)\cdot 37^{2} + \left(19 a + 19\right)\cdot 37^{3} + \left(18 a + 26\right)\cdot 37^{4} +O(37^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)$ | $1$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.