Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(393477257\)\(\medspace = 17^{3} \cdot 283^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.393477257.2 |
| 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 4.2.283.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 17x^{3} + 71x^{2} + 138x + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 5 a + 22 + \left(48 a + 34\right)\cdot 53 + \left(38 a + 39\right)\cdot 53^{2} + \left(49 a + 1\right)\cdot 53^{3} + \left(2 a + 42\right)\cdot 53^{4} + \left(3 a + 50\right)\cdot 53^{5} + 14 a\cdot 53^{6} + \left(8 a + 1\right)\cdot 53^{7} + \left(33 a + 17\right)\cdot 53^{8} + \left(38 a + 37\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 21\cdot 53 + 25\cdot 53^{2} + 4\cdot 53^{3} + 53^{4} + 32\cdot 53^{5} + 27\cdot 53^{6} + 25\cdot 53^{7} + 5\cdot 53^{8} + 44\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 27 + \left(17 a + 26\right)\cdot 53 + \left(17 a + 50\right)\cdot 53^{2} + \left(37 a + 41\right)\cdot 53^{3} + \left(a + 52\right)\cdot 53^{4} + \left(42 a + 40\right)\cdot 53^{5} + \left(36 a + 6\right)\cdot 53^{6} + \left(15 a + 21\right)\cdot 53^{7} + \left(12 a + 7\right)\cdot 53^{8} + \left(51 a + 6\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a + 43 + \left(35 a + 37\right)\cdot 53 + \left(35 a + 49\right)\cdot 53^{2} + \left(15 a + 14\right)\cdot 53^{3} + \left(51 a + 22\right)\cdot 53^{4} + \left(10 a + 48\right)\cdot 53^{5} + \left(16 a + 5\right)\cdot 53^{6} + \left(37 a + 47\right)\cdot 53^{7} + \left(40 a + 40\right)\cdot 53^{8} + \left(a + 39\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 48 a + 42 + \left(4 a + 9\right)\cdot 53 + \left(14 a + 41\right)\cdot 53^{2} + \left(3 a + 2\right)\cdot 53^{3} + \left(50 a + 4\right)\cdot 53^{4} + \left(49 a + 7\right)\cdot 53^{5} + \left(38 a + 1\right)\cdot 53^{6} + \left(44 a + 20\right)\cdot 53^{7} + \left(19 a + 35\right)\cdot 53^{8} + \left(14 a + 52\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 28\cdot 53 + 5\cdot 53^{2} + 40\cdot 53^{3} + 36\cdot 53^{4} + 32\cdot 53^{5} + 10\cdot 53^{6} + 44\cdot 53^{7} + 52\cdot 53^{8} + 31\cdot 53^{9} +O(53^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,5)$ | $-1$ |
| $6$ | $4$ | $(1,6,4,2)(3,5)$ | $1$ |
| $8$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.