Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(23488\)\(\medspace = 2^{6} \cdot 367 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.187904.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | odd |
| Determinant: | 1.367.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.4.187904.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 4x^{4} - 2x^{3} + 3x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 13 + \left(6 a + 16\right)\cdot 23 + \left(4 a + 20\right)\cdot 23^{2} + \left(13 a + 14\right)\cdot 23^{3} + \left(17 a + 2\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 7 + \left(22 a + 15\right)\cdot 23 + \left(2 a + 2\right)\cdot 23^{2} + \left(6 a + 9\right)\cdot 23^{3} + \left(22 a + 20\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 13 + 10\cdot 23 + \left(20 a + 9\right)\cdot 23^{2} + \left(16 a + 18\right)\cdot 23^{3} + 12\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 23 + 2\cdot 23^{2} + 3\cdot 23^{3} + 22\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 3\cdot 23 + 11\cdot 23^{2} + 9\cdot 23^{3} + 9\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 6 + \left(16 a + 21\right)\cdot 23 + \left(18 a + 22\right)\cdot 23^{2} + \left(9 a + 13\right)\cdot 23^{3} + \left(5 a + 1\right)\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.