Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(3003175\)\(\medspace = 5^{2} \cdot 7 \cdot 131^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.3003175.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.32095.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{4} + 3x^{3} + 3x^{2} - x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 9\cdot 23 + 19\cdot 23^{2} + 19\cdot 23^{3} + 13\cdot 23^{4} + 2\cdot 23^{5} + 15\cdot 23^{7} + 8\cdot 23^{8} + 2\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 11 + \left(9 a + 9\right)\cdot 23 + \left(3 a + 10\right)\cdot 23^{2} + \left(16 a + 16\right)\cdot 23^{3} + \left(8 a + 2\right)\cdot 23^{4} + \left(12 a + 18\right)\cdot 23^{5} + \left(17 a + 5\right)\cdot 23^{6} + \left(22 a + 17\right)\cdot 23^{7} + \left(10 a + 9\right)\cdot 23^{8} + \left(15 a + 6\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 9 + \left(8 a + 5\right)\cdot 23 + 15\cdot 23^{2} + \left(17 a + 5\right)\cdot 23^{3} + \left(5 a + 3\right)\cdot 23^{4} + \left(12 a + 19\right)\cdot 23^{5} + \left(12 a + 20\right)\cdot 23^{6} + \left(20 a + 9\right)\cdot 23^{7} + \left(3 a + 22\right)\cdot 23^{8} + \left(18 a + 10\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 21 + \left(14 a + 15\right)\cdot 23 + \left(22 a + 7\right)\cdot 23^{2} + \left(5 a + 16\right)\cdot 23^{3} + \left(17 a + 20\right)\cdot 23^{4} + \left(10 a + 14\right)\cdot 23^{5} + \left(10 a + 10\right)\cdot 23^{6} + \left(2 a + 15\right)\cdot 23^{7} + \left(19 a + 9\right)\cdot 23^{8} + \left(4 a + 20\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 a + 15 + \left(13 a + 2\right)\cdot 23 + \left(19 a + 8\right)\cdot 23^{2} + \left(6 a + 22\right)\cdot 23^{3} + \left(14 a + 3\right)\cdot 23^{4} + \left(10 a + 11\right)\cdot 23^{5} + \left(5 a + 5\right)\cdot 23^{6} + 22\cdot 23^{7} + \left(12 a + 8\right)\cdot 23^{8} + \left(7 a + 3\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 + 3\cdot 23 + 8\cdot 23^{2} + 11\cdot 23^{3} + 23^{4} + 3\cdot 23^{5} + 3\cdot 23^{6} + 12\cdot 23^{7} + 9\cdot 23^{8} + 2\cdot 23^{9} +O(23^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $-1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $1$ |
| $8$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.