Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(33856\)\(\medspace = 2^{6} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.33856.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1472.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 50 a + 8 + \left(30 a + 19\right)\cdot 53 + \left(24 a + 47\right)\cdot 53^{2} + \left(21 a + 12\right)\cdot 53^{3} + \left(39 a + 17\right)\cdot 53^{4} + 7\cdot 53^{5} + \left(13 a + 46\right)\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 50 a + 4 + \left(30 a + 13\right)\cdot 53 + \left(24 a + 44\right)\cdot 53^{2} + \left(21 a + 31\right)\cdot 53^{3} + \left(39 a + 5\right)\cdot 53^{4} + 29\cdot 53^{5} + \left(13 a + 8\right)\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 33\cdot 53 + 46\cdot 53^{2} + 34\cdot 53^{3} + 27\cdot 53^{4} + 50\cdot 53^{5} + 4\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 45 + \left(22 a + 33\right)\cdot 53 + \left(28 a + 5\right)\cdot 53^{2} + \left(31 a + 40\right)\cdot 53^{3} + \left(13 a + 35\right)\cdot 53^{4} + \left(52 a + 45\right)\cdot 53^{5} + \left(39 a + 6\right)\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 49 + \left(22 a + 39\right)\cdot 53 + \left(28 a + 8\right)\cdot 53^{2} + \left(31 a + 21\right)\cdot 53^{3} + \left(13 a + 47\right)\cdot 53^{4} + \left(52 a + 23\right)\cdot 53^{5} + \left(39 a + 44\right)\cdot 53^{6} +O(53^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 + 19\cdot 53 + 6\cdot 53^{2} + 18\cdot 53^{3} + 25\cdot 53^{4} + 2\cdot 53^{5} + 48\cdot 53^{6} +O(53^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)$ | $-1$ |
| $6$ | $4$ | $(1,6,4,3)(2,5)$ | $1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.