Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(1560895637\)\(\medspace = 7^{2} \cdot 317^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.222985091.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.317.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.4.222985091.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 21x^{3} - 18x^{2} - 19x + 11 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 21 a + 6 + \left(12 a + 7\right)\cdot 23 + \left(7 a + 1\right)\cdot 23^{2} + \left(22 a + 12\right)\cdot 23^{3} + \left(17 a + 2\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 4\cdot 23 + 10\cdot 23^{2} + 22\cdot 23^{3} + 22\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 2 + \left(10 a + 12\right)\cdot 23 + \left(15 a + 3\right)\cdot 23^{2} + 3\cdot 23^{3} + \left(5 a + 16\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 3\cdot 23 + 18\cdot 23^{2} + 7\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 18 + \left(8 a + 19\right)\cdot 23 + \left(7 a + 14\right)\cdot 23^{2} + \left(17 a + 9\right)\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 2 + \left(14 a + 22\right)\cdot 23 + \left(15 a + 20\right)\cdot 23^{2} + \left(5 a + 13\right)\cdot 23^{3} + \left(22 a + 3\right)\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(1,3)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $1$ |
| $4$ | $3$ | $(1,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $1$ |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.