Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(1078727\)\(\medspace = 13^{3} \cdot 491 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.529654957.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.6383.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.491.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 8x^{4} + 44x^{3} - 14x^{2} - 43x - 35 \)
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The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 38 + \left(2 a + 3\right)\cdot 53 + \left(24 a + 13\right)\cdot 53^{2} + \left(35 a + 5\right)\cdot 53^{3} + \left(45 a + 13\right)\cdot 53^{4} + \left(43 a + 33\right)\cdot 53^{5} + \left(38 a + 19\right)\cdot 53^{6} + \left(34 a + 45\right)\cdot 53^{7} + \left(52 a + 36\right)\cdot 53^{8} + \left(34 a + 46\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 a + 37 + \left(50 a + 52\right)\cdot 53 + 28 a\cdot 53^{2} + \left(17 a + 17\right)\cdot 53^{3} + \left(7 a + 1\right)\cdot 53^{4} + \left(9 a + 4\right)\cdot 53^{5} + \left(14 a + 25\right)\cdot 53^{6} + \left(18 a + 39\right)\cdot 53^{7} + \left(18 a + 28\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 + 37\cdot 53 + 7\cdot 53^{2} + 41\cdot 53^{3} + 3\cdot 53^{4} + 4\cdot 53^{5} + 40\cdot 53^{6} + 17\cdot 53^{7} + 14\cdot 53^{8} + 36\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 a + 28 + \left(43 a + 15\right)\cdot 53 + \left(49 a + 48\right)\cdot 53^{2} + \left(6 a + 35\right)\cdot 53^{3} + \left(42 a + 43\right)\cdot 53^{4} + \left(16 a + 15\right)\cdot 53^{5} + \left(40 a + 21\right)\cdot 53^{6} + \left(28 a + 45\right)\cdot 53^{7} + \left(8 a + 20\right)\cdot 53^{8} + \left(40 a + 43\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 49\cdot 53 + 43\cdot 53^{2} + 45\cdot 53^{3} + 50\cdot 53^{4} + 7\cdot 53^{5} + 46\cdot 53^{6} + 49\cdot 53^{7} + 6\cdot 53^{8} + 21\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 46 + \left(9 a + 52\right)\cdot 53 + \left(3 a + 44\right)\cdot 53^{2} + \left(46 a + 13\right)\cdot 53^{3} + \left(10 a + 46\right)\cdot 53^{4} + \left(36 a + 40\right)\cdot 53^{5} + \left(12 a + 6\right)\cdot 53^{6} + \left(24 a + 14\right)\cdot 53^{7} + \left(44 a + 26\right)\cdot 53^{8} + \left(12 a + 36\right)\cdot 53^{9} +O(53^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,6,5)$ | $1$ |
| $6$ | $4$ | $(1,4,6,2)(3,5)$ | $-1$ |
| $8$ | $6$ | $(1,2,3,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.