Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(1000188\)\(\medspace = 2^{2} \cdot 3^{6} \cdot 7^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.7001316.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.28.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.7001316.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{4} - 3x^{3} + 9x^{2} + 9x + 4 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 22 a + 36 + \left(35 a + 8\right)\cdot 37 + \left(11 a + 7\right)\cdot 37^{2} + \left(29 a + 20\right)\cdot 37^{3} + \left(30 a + 9\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 9\cdot 37 + 10\cdot 37^{2} + 2\cdot 37^{3} + 35\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 13 + \left(a + 18\right)\cdot 37 + \left(25 a + 19\right)\cdot 37^{2} + \left(7 a + 14\right)\cdot 37^{3} + \left(6 a + 29\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 28 + \left(31 a + 1\right)\cdot 37 + \left(23 a + 6\right)\cdot 37^{2} + \left(35 a + 4\right)\cdot 37^{3} + \left(3 a + 25\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 17 + \left(5 a + 29\right)\cdot 37 + \left(13 a + 32\right)\cdot 37^{2} + \left(a + 11\right)\cdot 37^{3} + \left(33 a + 5\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 5\cdot 37 + 35\cdot 37^{2} + 20\cdot 37^{3} + 6\cdot 37^{4} +O(37^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $6$ | $2$ | $(1,2)$ | $2$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $12$ | $6$ | $(1,2)(4,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.