Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(912\)\(\medspace = 2^{4} \cdot 3 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.69312.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.57.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2736.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + x^{4} - 2x^{3} + x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 51 a + 62 + \left(31 a + 21\right)\cdot 67 + \left(28 a + 32\right)\cdot 67^{2} + \left(14 a + 59\right)\cdot 67^{3} + \left(11 a + 63\right)\cdot 67^{4} + \left(63 a + 35\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 3\cdot 67 + 4\cdot 67^{2} + 35\cdot 67^{3} + 47\cdot 67^{4} + 46\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 a + 53 + \left(33 a + 40\right)\cdot 67 + \left(41 a + 13\right)\cdot 67^{2} + \left(52 a + 57\right)\cdot 67^{3} + \left(10 a + 4\right)\cdot 67^{4} + \left(22 a + 49\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 65 + \left(35 a + 30\right)\cdot 67 + \left(38 a + 47\right)\cdot 67^{2} + \left(52 a + 21\right)\cdot 67^{3} + \left(55 a + 27\right)\cdot 67^{4} + \left(3 a + 9\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 44\cdot 67 + 24\cdot 67^{2} + 2\cdot 67^{3} + 62\cdot 67^{4} + 66\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a + 48 + \left(33 a + 59\right)\cdot 67 + \left(25 a + 11\right)\cdot 67^{2} + \left(14 a + 25\right)\cdot 67^{3} + \left(56 a + 62\right)\cdot 67^{4} + \left(44 a + 59\right)\cdot 67^{5} +O(67^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.