Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(4255443\)\(\medspace = 3^{3} \cdot 397^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.4255443.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.10719.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 7x^{4} + 6x^{2} + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a + 4 + \left(2 a + 3\right)\cdot 7 + \left(6 a + 6\right)\cdot 7^{2} + \left(4 a + 6\right)\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} + \left(2 a + 1\right)\cdot 7^{5} + \left(2 a + 4\right)\cdot 7^{6} + \left(a + 2\right)\cdot 7^{7} + \left(2 a + 3\right)\cdot 7^{8} + \left(4 a + 1\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 7 + 5\cdot 7^{2} + 5\cdot 7^{4} + 2\cdot 7^{5} + 2\cdot 7^{6} + 3\cdot 7^{7} + 3\cdot 7^{8} + 7^{9} +O(7^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 2 + \left(4 a + 1\right)\cdot 7 + 3\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + 3\cdot 7^{4} + \left(4 a + 4\right)\cdot 7^{5} + \left(4 a + 3\right)\cdot 7^{6} + \left(5 a + 1\right)\cdot 7^{7} + \left(4 a + 4\right)\cdot 7^{8} + \left(2 a + 3\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 3 + \left(4 a + 3\right)\cdot 7 + 2 a\cdot 7^{3} + 5\cdot 7^{4} + \left(4 a + 5\right)\cdot 7^{5} + \left(4 a + 2\right)\cdot 7^{6} + \left(5 a + 4\right)\cdot 7^{7} + \left(4 a + 3\right)\cdot 7^{8} + \left(2 a + 5\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 5\cdot 7 + 7^{2} + 6\cdot 7^{3} + 7^{4} + 4\cdot 7^{5} + 4\cdot 7^{6} + 3\cdot 7^{7} + 3\cdot 7^{8} + 5\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 5 + \left(2 a + 5\right)\cdot 7 + \left(6 a + 3\right)\cdot 7^{2} + \left(4 a + 1\right)\cdot 7^{3} + \left(6 a + 3\right)\cdot 7^{4} + \left(2 a + 2\right)\cdot 7^{5} + \left(2 a + 3\right)\cdot 7^{6} + \left(a + 5\right)\cdot 7^{7} + \left(2 a + 2\right)\cdot 7^{8} + \left(4 a + 3\right)\cdot 7^{9} +O(7^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,6,5,3)$ | $1$ |
| $8$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.