Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(75272\)\(\medspace = 2^{3} \cdot 97^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.49664.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.49664.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 3x^{4} - 2x^{3} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 44\cdot 47 + 40\cdot 47^{2} + 2\cdot 47^{3} + 14\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 35 + \left(41 a + 3\right)\cdot 47 + \left(21 a + 27\right)\cdot 47^{2} + \left(5 a + 46\right)\cdot 47^{3} + \left(27 a + 40\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 19\cdot 47 + 13\cdot 47^{2} + 11\cdot 47^{3} + 9\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a + 41 + \left(5 a + 35\right)\cdot 47 + \left(25 a + 29\right)\cdot 47^{2} + \left(41 a + 35\right)\cdot 47^{3} + \left(19 a + 42\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 29 + \left(18 a + 32\right)\cdot 47 + \left(3 a + 20\right)\cdot 47^{2} + \left(34 a + 13\right)\cdot 47^{3} + \left(13 a + 20\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 16 + \left(28 a + 5\right)\cdot 47 + \left(43 a + 9\right)\cdot 47^{2} + \left(12 a + 31\right)\cdot 47^{3} + \left(33 a + 13\right)\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)$ | $-2$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,4,3)$ | $-1$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.