Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(19645443\)\(\medspace = 3^{3} \cdot 853^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.23031.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.23031.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + x^{4} - x^{3} + x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + \left(16 a + 7\right)\cdot 31 + \left(21 a + 8\right)\cdot 31^{2} + \left(4 a + 4\right)\cdot 31^{3} + \left(4 a + 8\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 21 + \left(14 a + 14\right)\cdot 31 + \left(9 a + 3\right)\cdot 31^{2} + \left(26 a + 23\right)\cdot 31^{3} + \left(26 a + 11\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 28\cdot 31 + 13\cdot 31^{2} + 9\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a + 6 + \left(12 a + 27\right)\cdot 31 + \left(3 a + 26\right)\cdot 31^{2} + \left(20 a + 7\right)\cdot 31^{3} + \left(2 a + 18\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 + 9\cdot 31 + 19\cdot 31^{2} + 3\cdot 31^{3} + 11\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 5 + \left(18 a + 6\right)\cdot 31 + \left(27 a + 21\right)\cdot 31^{2} + \left(10 a + 13\right)\cdot 31^{3} + \left(28 a + 3\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(2,5)$ | $0$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $4$ | $3$ | $(1,2,5)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,6,5,3)$ | $1$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.