Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(3984016\)\(\medspace = 2^{4} \cdot 499^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.3984016.1 |
| 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 4.2.31936.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 4x^{4} - 5x^{2} + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 7\cdot 13 + 10\cdot 13^{2} + 9\cdot 13^{3} + 13^{4} + 8\cdot 13^{6} + 4\cdot 13^{7} + 9\cdot 13^{8} + 2\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 5 + \left(a + 3\right)\cdot 13 + a\cdot 13^{2} + \left(3 a + 6\right)\cdot 13^{3} + \left(10 a + 2\right)\cdot 13^{4} + \left(12 a + 11\right)\cdot 13^{5} + 5 a\cdot 13^{6} + \left(a + 12\right)\cdot 13^{7} + 7 a\cdot 13^{8} + \left(11 a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 6 + \left(a + 10\right)\cdot 13 + \left(a + 12\right)\cdot 13^{2} + \left(3 a + 4\right)\cdot 13^{3} + \left(10 a + 3\right)\cdot 13^{4} + \left(12 a + 12\right)\cdot 13^{5} + \left(5 a + 5\right)\cdot 13^{6} + \left(a + 5\right)\cdot 13^{7} + \left(7 a + 6\right)\cdot 13^{8} + \left(11 a + 2\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 5\cdot 13 + 2\cdot 13^{2} + 3\cdot 13^{3} + 11\cdot 13^{4} + 12\cdot 13^{5} + 4\cdot 13^{6} + 8\cdot 13^{7} + 3\cdot 13^{8} + 10\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 8 + \left(11 a + 9\right)\cdot 13 + \left(11 a + 12\right)\cdot 13^{2} + \left(9 a + 6\right)\cdot 13^{3} + \left(2 a + 10\right)\cdot 13^{4} + 13^{5} + \left(7 a + 12\right)\cdot 13^{6} + 11 a\cdot 13^{7} + \left(5 a + 12\right)\cdot 13^{8} + \left(a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 7 + \left(11 a + 2\right)\cdot 13 + 11 a\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(2 a + 9\right)\cdot 13^{4} + \left(7 a + 7\right)\cdot 13^{6} + \left(11 a + 7\right)\cdot 13^{7} + \left(5 a + 6\right)\cdot 13^{8} + \left(a + 10\right)\cdot 13^{9} +O(13^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,5,4,2)(3,6)$ | $1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.