Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(10051\)\(\medspace = 19 \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.231173.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.8303.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 5x^{4} + 2x^{3} + 9x^{2} + 2x - 7 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 55 + \left(24 a + 24\right)\cdot 67 + \left(24 a + 6\right)\cdot 67^{2} + \left(38 a + 14\right)\cdot 67^{3} + \left(50 a + 64\right)\cdot 67^{4} + \left(34 a + 46\right)\cdot 67^{5} + \left(31 a + 19\right)\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a + 12 + \left(17 a + 66\right)\cdot 67 + \left(45 a + 22\right)\cdot 67^{2} + \left(46 a + 23\right)\cdot 67^{3} + \left(57 a + 59\right)\cdot 67^{4} + \left(65 a + 15\right)\cdot 67^{5} + \left(20 a + 50\right)\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 a + 45 + \left(49 a + 43\right)\cdot 67 + \left(21 a + 52\right)\cdot 67^{2} + \left(20 a + 30\right)\cdot 67^{3} + \left(9 a + 42\right)\cdot 67^{4} + \left(a + 20\right)\cdot 67^{5} + \left(46 a + 1\right)\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 a + 4 + \left(42 a + 50\right)\cdot 67 + \left(42 a + 12\right)\cdot 67^{2} + \left(28 a + 9\right)\cdot 67^{3} + \left(16 a + 27\right)\cdot 67^{4} + \left(32 a + 1\right)\cdot 67^{5} + \left(35 a + 44\right)\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 + 40\cdot 67 + 54\cdot 67^{2} + 46\cdot 67^{3} + 63\cdot 67^{4} + 14\cdot 67^{5} + 21\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 39 + 42\cdot 67 + 51\cdot 67^{2} + 9\cdot 67^{3} + 11\cdot 67^{4} + 34\cdot 67^{5} + 64\cdot 67^{6} +O(67^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(3,5)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(1,4,2,3)$ | $1$ |
| $6$ | $4$ | $(1,6,2,5)(3,4)$ | $-1$ |
| $8$ | $6$ | $(1,4,6,2,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.