Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(41328\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.142333632.5 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.1148.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.13776.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 7x^{4} + 4x^{2} + 12 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 19 + \left(13 a + 23\right)\cdot 31 + 21\cdot 31^{2} + \left(14 a + 1\right)\cdot 31^{3} + \left(14 a + 8\right)\cdot 31^{4} + \left(15 a + 7\right)\cdot 31^{5} + \left(30 a + 8\right)\cdot 31^{6} + \left(4 a + 10\right)\cdot 31^{7} + 2 a\cdot 31^{8} + \left(22 a + 10\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 16 + \left(a + 21\right)\cdot 31 + \left(28 a + 3\right)\cdot 31^{2} + \left(21 a + 23\right)\cdot 31^{3} + \left(24 a + 1\right)\cdot 31^{4} + \left(7 a + 20\right)\cdot 31^{5} + \left(15 a + 19\right)\cdot 31^{6} + \left(21 a + 1\right)\cdot 31^{7} + \left(25 a + 16\right)\cdot 31^{8} + \left(11 a + 16\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 13\cdot 31 + 20\cdot 31^{2} + 9\cdot 31^{3} + 24\cdot 31^{4} + 13\cdot 31^{5} + 13\cdot 31^{6} + 26\cdot 31^{7} + 2\cdot 31^{8} + 22\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 a + 12 + \left(17 a + 7\right)\cdot 31 + \left(30 a + 9\right)\cdot 31^{2} + \left(16 a + 29\right)\cdot 31^{3} + \left(16 a + 22\right)\cdot 31^{4} + \left(15 a + 23\right)\cdot 31^{5} + 22\cdot 31^{6} + \left(26 a + 20\right)\cdot 31^{7} + \left(28 a + 30\right)\cdot 31^{8} + \left(8 a + 20\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 15 + \left(29 a + 9\right)\cdot 31 + \left(2 a + 27\right)\cdot 31^{2} + \left(9 a + 7\right)\cdot 31^{3} + \left(6 a + 29\right)\cdot 31^{4} + \left(23 a + 10\right)\cdot 31^{5} + \left(15 a + 11\right)\cdot 31^{6} + \left(9 a + 29\right)\cdot 31^{7} + \left(5 a + 14\right)\cdot 31^{8} + \left(19 a + 14\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 17\cdot 31 + 10\cdot 31^{2} + 21\cdot 31^{3} + 6\cdot 31^{4} + 17\cdot 31^{5} + 17\cdot 31^{6} + 4\cdot 31^{7} + 28\cdot 31^{8} + 8\cdot 31^{9} +O(31^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(2,6,5,3)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,2,6,4,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.