Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(440657\)\(\medspace = 7^{2} \cdot 17 \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.440657.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.17.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.325703.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - x^{4} + 7x^{3} + 3x^{2} - 7x - 11 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{2} + 82x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 54 + 22\cdot 89 + 78\cdot 89^{2} + 63\cdot 89^{3} + 3\cdot 89^{4} + 38\cdot 89^{5} + 61\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 + 66\cdot 89 + 10\cdot 89^{2} + 25\cdot 89^{3} + 85\cdot 89^{4} + 50\cdot 89^{5} + 27\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 a + 68 + \left(57 a + 77\right)\cdot 89 + \left(9 a + 12\right)\cdot 89^{2} + \left(42 a + 59\right)\cdot 89^{3} + \left(17 a + 36\right)\cdot 89^{4} + \left(44 a + 57\right)\cdot 89^{5} + \left(30 a + 72\right)\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 55 a + 82 + \left(57 a + 18\right)\cdot 89 + \left(9 a + 66\right)\cdot 89^{2} + \left(42 a + 11\right)\cdot 89^{3} + \left(17 a + 61\right)\cdot 89^{4} + \left(44 a + 6\right)\cdot 89^{5} + \left(30 a + 25\right)\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a + 22 + \left(31 a + 11\right)\cdot 89 + \left(79 a + 76\right)\cdot 89^{2} + \left(46 a + 29\right)\cdot 89^{3} + \left(71 a + 52\right)\cdot 89^{4} + \left(44 a + 31\right)\cdot 89^{5} + \left(58 a + 16\right)\cdot 89^{6} +O(89^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 a + 8 + \left(31 a + 70\right)\cdot 89 + \left(79 a + 22\right)\cdot 89^{2} + \left(46 a + 77\right)\cdot 89^{3} + \left(71 a + 27\right)\cdot 89^{4} + \left(44 a + 82\right)\cdot 89^{5} + \left(58 a + 63\right)\cdot 89^{6} +O(89^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,6)(3,5)$ | $1$ |
| $8$ | $3$ | $(1,4,3)(2,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,2,5)$ | $-1$ |
| $6$ | $4$ | $(1,6,2,4)(3,5)$ | $1$ |
| $8$ | $6$ | $(1,4,3,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.