Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(991\!\cdots\!375\)\(\medspace = 5^{9} \cdot 7^{9} \cdot 13^{9} \cdot 17^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.83490514835.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.7735.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.83490514835.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 3x^{6} + 6x^{5} + 13x^{4} - 15x^{3} - 21x^{2} + 11x - 25 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 14 + 6\cdot 53 + 42\cdot 53^{2} + 38\cdot 53^{3} + 49\cdot 53^{4} + 14\cdot 53^{5} + 37\cdot 53^{6} + 14\cdot 53^{7} + 16\cdot 53^{8} + 45\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 2 + \left(49 a + 29\right)\cdot 53 + \left(22 a + 29\right)\cdot 53^{2} + \left(22 a + 50\right)\cdot 53^{3} + \left(12 a + 43\right)\cdot 53^{4} + \left(27 a + 5\right)\cdot 53^{5} + \left(33 a + 49\right)\cdot 53^{6} + \left(51 a + 19\right)\cdot 53^{7} + \left(3 a + 2\right)\cdot 53^{8} + \left(7 a + 33\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 22 + \left(15 a + 6\right)\cdot 53 + \left(24 a + 48\right)\cdot 53^{2} + \left(a + 14\right)\cdot 53^{3} + \left(22 a + 26\right)\cdot 53^{4} + \left(35 a + 13\right)\cdot 53^{5} + \left(25 a + 41\right)\cdot 53^{6} + \left(46 a + 22\right)\cdot 53^{7} + \left(4 a + 27\right)\cdot 53^{8} + \left(9 a + 3\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 a + 39 + \left(46 a + 40\right)\cdot 53 + \left(21 a + 34\right)\cdot 53^{2} + \left(11 a + 36\right)\cdot 53^{3} + \left(7 a + 39\right)\cdot 53^{4} + \left(50 a + 34\right)\cdot 53^{5} + \left(22 a + 35\right)\cdot 53^{6} + \left(37 a + 15\right)\cdot 53^{7} + \left(31 a + 50\right)\cdot 53^{8} + \left(12 a + 24\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 4 + \left(6 a + 37\right)\cdot 53 + \left(31 a + 22\right)\cdot 53^{2} + \left(41 a + 7\right)\cdot 53^{3} + \left(45 a + 4\right)\cdot 53^{4} + \left(2 a + 16\right)\cdot 53^{5} + \left(30 a + 24\right)\cdot 53^{6} + \left(15 a + 36\right)\cdot 53^{7} + \left(21 a + 33\right)\cdot 53^{8} + \left(40 a + 43\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 48 + \left(37 a + 35\right)\cdot 53 + \left(28 a + 23\right)\cdot 53^{2} + \left(51 a + 49\right)\cdot 53^{3} + \left(30 a + 6\right)\cdot 53^{4} + \left(17 a + 27\right)\cdot 53^{5} + \left(27 a + 2\right)\cdot 53^{6} + \left(6 a + 24\right)\cdot 53^{7} + 48 a\cdot 53^{8} + \left(43 a + 35\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 + 4\cdot 53 + 45\cdot 53^{2} + 2\cdot 53^{3} + 23\cdot 53^{4} + 50\cdot 53^{5} + 24\cdot 53^{6} + 44\cdot 53^{7} + 8\cdot 53^{8} + 22\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 38 a + 9 + \left(3 a + 52\right)\cdot 53 + \left(30 a + 18\right)\cdot 53^{2} + \left(30 a + 11\right)\cdot 53^{3} + \left(40 a + 18\right)\cdot 53^{4} + \left(25 a + 49\right)\cdot 53^{5} + \left(19 a + 49\right)\cdot 53^{6} + \left(a + 33\right)\cdot 53^{7} + \left(49 a + 19\right)\cdot 53^{8} + \left(45 a + 4\right)\cdot 53^{9} +O(53^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,6)(3,7)$ | $-6$ |
| $9$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,5)(4,8)$ | $0$ |
| $16$ | $3$ | $(3,6,7)$ | $0$ |
| $64$ | $3$ | $(3,6,7)(4,5,8)$ | $0$ |
| $12$ | $4$ | $(1,3,6,7)$ | $0$ |
| $36$ | $4$ | $(1,3,6,7)(2,4,5,8)$ | $-2$ |
| $36$ | $4$ | $(1,6)(2,4,5,8)(3,7)$ | $0$ |
| $72$ | $4$ | $(1,5,6,2)(3,8,7,4)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,4,5,8)$ | $2$ |
| $144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(2,5)(3,7,6)(4,8)$ | $0$ |
| $96$ | $6$ | $(2,4)(3,6,7)$ | $0$ |
| $192$ | $6$ | $(1,2)(3,5,6,8,7,4)$ | $0$ |
| $144$ | $8$ | $(1,4,3,5,6,8,7,2)$ | $0$ |
| $96$ | $12$ | $(2,4,5,8)(3,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.