Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(717\)\(\medspace = 3 \cdot 239 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.171363.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.717.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2151.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 2x^{4} - 3x^{3} + 2x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 41 + 39\cdot 43 + 43^{2} + 16\cdot 43^{3} + 40\cdot 43^{4} + 5\cdot 43^{5} + 17\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 18 + \left(18 a + 21\right)\cdot 43 + \left(35 a + 41\right)\cdot 43^{2} + \left(24 a + 8\right)\cdot 43^{3} + \left(37 a + 32\right)\cdot 43^{4} + \left(36 a + 42\right)\cdot 43^{5} + \left(31 a + 10\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 a + 26 + \left(24 a + 31\right)\cdot 43 + \left(7 a + 15\right)\cdot 43^{2} + \left(18 a + 41\right)\cdot 43^{3} + \left(5 a + 1\right)\cdot 43^{4} + \left(6 a + 42\right)\cdot 43^{5} + \left(11 a + 5\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 27 + \left(17 a + 9\right)\cdot 43 + \left(15 a + 23\right)\cdot 43^{2} + \left(26 a + 19\right)\cdot 43^{3} + \left(41 a + 18\right)\cdot 43^{4} + \left(9 a + 35\right)\cdot 43^{5} + \left(28 a + 16\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 a + 39 + \left(25 a + 14\right)\cdot 43 + \left(27 a + 21\right)\cdot 43^{2} + \left(16 a + 30\right)\cdot 43^{3} + \left(a + 33\right)\cdot 43^{4} + \left(33 a + 3\right)\cdot 43^{5} + \left(14 a + 35\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 11\cdot 43 + 25\cdot 43^{2} + 12\cdot 43^{3} + 2\cdot 43^{4} + 42\cdot 43^{5} + 42\cdot 43^{6} +O(43^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,4,6,3)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.