Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(556\!\cdots\!424\)\(\medspace = 2^{53} \cdot 3^{31} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.7739670528.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.24.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.7739670528.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - 4x^{5} - 6x^{4} - 24x^{3} + 10x^{2} + 12x + 9 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a^{2} + 7 a + 6 + \left(4 a^{2} + 6 a + 1\right)\cdot 11 + \left(5 a^{2} + 10 a + 6\right)\cdot 11^{2} + \left(4 a^{2} + 10 a\right)\cdot 11^{3} + \left(6 a^{2} + 7 a + 4\right)\cdot 11^{4} + \left(8 a^{2} + a + 7\right)\cdot 11^{5} + 9\cdot 11^{6} + \left(4 a^{2} + 4 a + 5\right)\cdot 11^{7} + \left(3 a^{2} + 4 a + 5\right)\cdot 11^{8} + \left(3 a + 2\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 5 + \left(9 a^{2} + 9 a + 3\right)\cdot 11 + \left(4 a^{2} + a + 5\right)\cdot 11^{2} + \left(4 a^{2} + 9 a\right)\cdot 11^{3} + \left(6 a^{2} + 8 a + 4\right)\cdot 11^{4} + \left(3 a^{2} + 2 a + 4\right)\cdot 11^{5} + \left(10 a + 5\right)\cdot 11^{6} + \left(8 a^{2} + a + 7\right)\cdot 11^{7} + \left(8 a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(4 a^{2} + 3 a + 1\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a^{2} + 2 a + 4 + \left(8 a^{2} + 6 a + 2\right)\cdot 11 + \left(9 a + 7\right)\cdot 11^{2} + \left(2 a^{2} + a + 4\right)\cdot 11^{3} + \left(9 a^{2} + 5 a\right)\cdot 11^{4} + \left(9 a^{2} + 6 a + 9\right)\cdot 11^{5} + \left(9 a^{2} + 10\right)\cdot 11^{6} + \left(9 a^{2} + 5 a + 9\right)\cdot 11^{7} + \left(9 a^{2} + 10 a + 6\right)\cdot 11^{8} + \left(5 a^{2} + 3 a + 2\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a^{2} + 5 a + 1 + \left(a^{2} + 8 a + 9\right)\cdot 11 + \left(3 a^{2} + 6 a + 1\right)\cdot 11^{2} + \left(2 a^{2} + 6 a + 6\right)\cdot 11^{3} + \left(3 a + 3\right)\cdot 11^{4} + \left(7 a^{2} + 4 a + 8\right)\cdot 11^{5} + \left(2 a^{2} + 5 a + 5\right)\cdot 11^{6} + \left(2 a^{2} + 5 a + 7\right)\cdot 11^{7} + \left(9 a^{2} + 9 a\right)\cdot 11^{8} + \left(a^{2} + 5 a + 6\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 + 10\cdot 11 + 6\cdot 11^{2} + 11^{3} + 11^{4} + 3\cdot 11^{5} + 4\cdot 11^{6} + 8\cdot 11^{7} + 11^{8} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 7 a + 3 + \left(6 a^{2} + 5 a + 9\right)\cdot 11 + \left(4 a + 5\right)\cdot 11^{2} + \left(8 a^{2} + 9 a + 6\right)\cdot 11^{3} + \left(3 a^{2} + 7 a + 4\right)\cdot 11^{4} + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{5} + \left(7 a^{2} + 5 a + 5\right)\cdot 11^{6} + \left(2 a^{2} + 4 a + 4\right)\cdot 11^{7} + \left(8 a^{2} + 3 a + 10\right)\cdot 11^{8} + \left(a^{2} + 5\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 + 3\cdot 11 + 3\cdot 11^{2} + 5\cdot 11^{3} + 2\cdot 11^{4} + 11^{5} + 7\cdot 11^{6} + 9\cdot 11^{7} + 7\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 a^{2} + 10 a + 9 + \left(2 a^{2} + 7 a + 3\right)\cdot 11 + \left(7 a^{2} + 10 a + 7\right)\cdot 11^{2} + \left(5 a + 7\right)\cdot 11^{3} + \left(7 a^{2} + 10 a + 1\right)\cdot 11^{4} + \left(7 a^{2} + a + 9\right)\cdot 11^{5} + 6\cdot 11^{6} + \left(6 a^{2} + a + 1\right)\cdot 11^{7} + \left(4 a^{2} + 9 a + 9\right)\cdot 11^{8} + \left(7 a^{2} + 4 a + 9\right)\cdot 11^{9} +O(11^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(4,6)(5,8)$ | $-6$ |
| $9$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,2)$ | $0$ |
| $24$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,2)(4,5)$ | $-2$ |
| $36$ | $2$ | $(1,2)(4,6)(5,8)$ | $0$ |
| $16$ | $3$ | $(1,3,7)$ | $0$ |
| $64$ | $3$ | $(1,3,7)(5,6,8)$ | $0$ |
| $12$ | $4$ | $(4,5,6,8)$ | $0$ |
| $36$ | $4$ | $(1,2,3,7)(4,5,6,8)$ | $-2$ |
| $36$ | $4$ | $(1,2,3,7)(4,6)(5,8)$ | $0$ |
| $72$ | $4$ | $(1,4,3,6)(2,5,7,8)$ | $0$ |
| $72$ | $4$ | $(1,2)(4,5,6,8)$ | $2$ |
| $144$ | $4$ | $(1,5,2,4)(3,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,7,3)(4,6)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,2)(5,8,6)$ | $0$ |
| $192$ | $6$ | $(1,5,3,6,7,8)(2,4)$ | $0$ |
| $144$ | $8$ | $(1,4,2,5,3,6,7,8)$ | $0$ |
| $96$ | $12$ | $(1,3,7)(4,5,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.