Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(960596\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 13^{2} \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.461634992.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.29.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.4.461634992.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 10x^{4} - 4x^{3} - 4x^{2} + 20x + 4 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 18 a + 15 + \left(17 a + 15\right)\cdot 23 + \left(7 a + 4\right)\cdot 23^{2} + \left(6 a + 20\right)\cdot 23^{3} + \left(16 a + 1\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 14\cdot 23 + 13\cdot 23^{2} + 9\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 11 + \left(16 a + 14\right)\cdot 23 + \left(2 a + 21\right)\cdot 23^{2} + \left(10 a + 20\right)\cdot 23^{3} + \left(a + 1\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 5 + \left(5 a + 10\right)\cdot 23 + \left(15 a + 2\right)\cdot 23^{2} + \left(16 a + 2\right)\cdot 23^{3} + \left(6 a + 5\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 + 20\cdot 23 + 15\cdot 23^{2} + 16\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 2 + \left(6 a + 17\right)\cdot 23 + \left(20 a + 10\right)\cdot 23^{2} + \left(12 a + 15\right)\cdot 23^{3} + \left(21 a + 17\right)\cdot 23^{4} +O(23^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(1,4)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $4$ | $3$ | $(2,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,3,4,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,5,6)$ | $-1$ |
| $12$ | $6$ | $(1,4)(2,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.