Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(38509088\)\(\medspace = 2^{5} \cdot 1097^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.2246656.1 |
| 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 6.2.2246656.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 7x^{4} - 8x^{3} + 9x^{2} - 6x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 7 a + 22 + \left(14 a + 14\right)\cdot 23 + \left(20 a + 14\right)\cdot 23^{2} + 11\cdot 23^{3} + \left(15 a + 9\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 9 + \left(16 a + 5\right)\cdot 23 + \left(14 a + 17\right)\cdot 23^{2} + \left(11 a + 22\right)\cdot 23^{3} + \left(11 a + 11\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 13 + \left(8 a + 13\right)\cdot 23 + \left(2 a + 18\right)\cdot 23^{2} + \left(22 a + 15\right)\cdot 23^{3} + \left(7 a + 15\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 + 10\cdot 23 + 21\cdot 23^{2} + 14\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 2 + \left(6 a + 7\right)\cdot 23 + \left(8 a + 7\right)\cdot 23^{2} + \left(11 a + 8\right)\cdot 23^{3} + 11 a\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 + 17\cdot 23 + 12\cdot 23^{2} + 18\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $4$ | $3$ | $(1,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,3,6)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.