Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(928\!\cdots\!504\)\(\medspace = 2^{36} \cdot 3^{38} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.11609505792.8 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.11609505792.8 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{5} + 18x^{4} - 24x^{3} + 40x^{2} - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{3} + 6x + 35 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{2} + 19 + \left(28 a^{2} + 8 a + 21\right)\cdot 37 + \left(6 a^{2} + 24 a + 18\right)\cdot 37^{2} + \left(10 a^{2} + 19 a + 16\right)\cdot 37^{3} + \left(35 a^{2} + 10 a + 13\right)\cdot 37^{4} + \left(22 a^{2} + 32 a + 10\right)\cdot 37^{5} + \left(33 a^{2} + 19 a + 34\right)\cdot 37^{6} + \left(27 a^{2} + 24 a + 3\right)\cdot 37^{7} + \left(35 a^{2} + 28 a + 35\right)\cdot 37^{8} + \left(34 a^{2} + 19 a + 25\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a^{2} + 35 a + 16 + \left(6 a^{2} + 20 a + 26\right)\cdot 37 + \left(18 a^{2} + a + 31\right)\cdot 37^{2} + \left(8 a^{2} + 34 a + 25\right)\cdot 37^{3} + \left(7 a^{2} + 23 a + 25\right)\cdot 37^{4} + \left(7 a^{2} + 24 a + 1\right)\cdot 37^{5} + \left(35 a^{2} + 21 a + 24\right)\cdot 37^{6} + \left(30 a^{2} + 13 a + 14\right)\cdot 37^{7} + \left(12 a^{2} + 5 a + 21\right)\cdot 37^{8} + \left(21 a^{2} + 28 a + 30\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 21\cdot 37 + 32\cdot 37^{2} + 35\cdot 37^{3} + 21\cdot 37^{4} + 11\cdot 37^{5} + 37^{6} + 2\cdot 37^{7} + 20\cdot 37^{8} +O(37^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a^{2} + 17 a + 14 + \left(34 a^{2} + 31 a + 29\right)\cdot 37 + \left(2 a^{2} + 18 a + 7\right)\cdot 37^{2} + \left(20 a^{2} + 27 a + 35\right)\cdot 37^{3} + \left(9 a^{2} + 24 a + 34\right)\cdot 37^{4} + \left(27 a^{2} + 5 a + 7\right)\cdot 37^{5} + \left(a^{2} + 7 a + 1\right)\cdot 37^{6} + \left(33 a^{2} + 12 a + 23\right)\cdot 37^{7} + \left(17 a^{2} + 34 a + 4\right)\cdot 37^{8} + \left(5 a^{2} + 27 a + 4\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 + 22\cdot 37 + 4\cdot 37^{2} + 24\cdot 37^{3} + 18\cdot 37^{5} + 20\cdot 37^{6} + 19\cdot 37^{7} + 23\cdot 37^{8} + 24\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a^{2} + 32 a + 6 + \left(32 a^{2} + 18 a + 1\right)\cdot 37 + \left(17 a^{2} + 30 a + 26\right)\cdot 37^{2} + \left(16 a^{2} + 2 a + 4\right)\cdot 37^{3} + \left(35 a^{2} + 23 a + 14\right)\cdot 37^{4} + \left(14 a^{2} + 18 a + 15\right)\cdot 37^{5} + \left(33 a^{2} + 18 a + 33\right)\cdot 37^{6} + \left(31 a^{2} + 6 a + 19\right)\cdot 37^{7} + \left(15 a^{2} + 2 a + 29\right)\cdot 37^{8} + \left(31 a^{2} + 16 a + 11\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 15 a^{2} + 22 a + 31 + \left(33 a^{2} + 21 a + 24\right)\cdot 37 + \left(15 a^{2} + 16 a + 22\right)\cdot 37^{2} + \left(8 a^{2} + 12 a + 25\right)\cdot 37^{3} + \left(20 a^{2} + 25 a + 3\right)\cdot 37^{4} + \left(2 a^{2} + 6 a + 20\right)\cdot 37^{5} + \left(8 a + 31\right)\cdot 37^{6} + \left(10 a^{2} + 11 a + 4\right)\cdot 37^{7} + \left(6 a^{2} + 34 a + 32\right)\cdot 37^{8} + \left(10 a^{2} + 17 a + 22\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 25 a^{2} + 5 a + 33 + \left(13 a^{2} + 10 a\right)\cdot 37 + \left(12 a^{2} + 19 a + 4\right)\cdot 37^{2} + \left(10 a^{2} + 14 a + 17\right)\cdot 37^{3} + \left(3 a^{2} + 3 a + 33\right)\cdot 37^{4} + \left(36 a^{2} + 23 a + 25\right)\cdot 37^{5} + \left(6 a^{2} + 35 a + 1\right)\cdot 37^{6} + \left(14 a^{2} + 5 a + 23\right)\cdot 37^{7} + \left(22 a^{2} + 6 a + 18\right)\cdot 37^{8} + \left(7 a^{2} + a + 27\right)\cdot 37^{9} +O(37^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,6)(3,8)$ | $-6$ |
| $9$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,5)(4,7)$ | $0$ |
| $16$ | $3$ | $(3,6,8)$ | $0$ |
| $64$ | $3$ | $(3,6,8)(4,5,7)$ | $0$ |
| $12$ | $4$ | $(1,3,6,8)$ | $0$ |
| $36$ | $4$ | $(1,3,6,8)(2,4,5,7)$ | $-2$ |
| $36$ | $4$ | $(1,6)(2,4,5,7)(3,8)$ | $0$ |
| $72$ | $4$ | $(1,5,6,2)(3,7,8,4)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,4,5,7)$ | $2$ |
| $144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(2,5)(3,8,6)(4,7)$ | $0$ |
| $96$ | $6$ | $(1,3)(4,7,5)$ | $0$ |
| $192$ | $6$ | $(1,2)(3,5,6,7,8,4)$ | $0$ |
| $144$ | $8$ | $(1,4,3,5,6,7,8,2)$ | $0$ |
| $96$ | $12$ | $(2,4,5,7)(3,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.