Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(90891328\)\(\medspace = 2^{6} \cdot 11^{4} \cdot 97 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.363565312.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.97.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.363565312.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} - 68x^{3} + 108x^{2} + 64x + 692 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 + 48\cdot 53 + 15\cdot 53^{2} + 39\cdot 53^{3} + 22\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 + 6\cdot 53 + 25\cdot 53^{2} + 17\cdot 53^{3} + 18\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 a + 18 + \left(19 a + 29\right)\cdot 53 + \left(47 a + 3\right)\cdot 53^{2} + 11\cdot 53^{3} + \left(40 a + 45\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 24 + \left(33 a + 26\right)\cdot 53 + \left(5 a + 14\right)\cdot 53^{2} + \left(52 a + 20\right)\cdot 53^{3} + \left(12 a + 45\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a + 11 + \left(24 a + 18\right)\cdot 53 + \left(11 a + 39\right)\cdot 53^{2} + \left(16 a + 8\right)\cdot 53^{3} + \left(26 a + 22\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 a + 41 + \left(28 a + 29\right)\cdot 53 + \left(41 a + 7\right)\cdot 53^{2} + \left(36 a + 9\right)\cdot 53^{3} + \left(26 a + 5\right)\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,5)$ | $2$ |
| $9$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $-2$ |
| $4$ | $3$ | $(2,3,4)$ | $1$ |
| $18$ | $4$ | $(1,3,5,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,2,5,3,6,4)$ | $0$ |
| $12$ | $6$ | $(1,5)(2,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.