Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(545\!\cdots\!000\)\(\medspace = 2^{40} \cdot 3^{26} \cdot 5^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.955514880.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.955514880.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 16x^{5} + 28x^{4} - 28x^{3} + 20x^{2} - 8x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{3} + x + 35 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a^{2} + 12 a + 38 + \left(4 a^{2} + 6 a + 7\right)\cdot 41 + \left(20 a + 19\right)\cdot 41^{2} + \left(18 a^{2} + 30 a + 22\right)\cdot 41^{3} + \left(14 a^{2} + 39 a + 8\right)\cdot 41^{4} + \left(15 a^{2} + 22 a + 25\right)\cdot 41^{5} + \left(9 a^{2} + 21 a + 1\right)\cdot 41^{6} + \left(5 a^{2} + 10 a + 11\right)\cdot 41^{7} + \left(17 a^{2} + 34 a + 5\right)\cdot 41^{8} + \left(6 a^{2} + 34 a + 36\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a^{2} + 23 a + 3 + \left(14 a^{2} + 35 a + 1\right)\cdot 41 + \left(10 a^{2} + 3 a + 26\right)\cdot 41^{2} + \left(20 a^{2} + 33 a + 37\right)\cdot 41^{3} + \left(31 a^{2} + 11 a + 19\right)\cdot 41^{4} + \left(9 a^{2} + 5 a + 21\right)\cdot 41^{5} + \left(4 a^{2} + 22 a + 25\right)\cdot 41^{6} + \left(21 a^{2} + 31 a + 21\right)\cdot 41^{7} + \left(37 a^{2} + 21 a + 32\right)\cdot 41^{8} + \left(7 a^{2} + 34 a + 9\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a^{2} + 3 a + 3 + \left(31 a^{2} + 23 a + 39\right)\cdot 41 + \left(28 a^{2} + 10 a + 7\right)\cdot 41^{2} + \left(21 a^{2} + 27 a + 25\right)\cdot 41^{3} + \left(32 a^{2} + 6 a + 26\right)\cdot 41^{4} + \left(15 a^{2} + 14 a + 33\right)\cdot 41^{5} + \left(35 a^{2} + 31 a + 32\right)\cdot 41^{6} + \left(14 a^{2} + 37 a + 12\right)\cdot 41^{7} + \left(31 a^{2} + 11 a + 36\right)\cdot 41^{8} + \left(6 a^{2} + 2 a + 40\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a^{2} + 6 a + 20 + \left(22 a^{2} + 40 a + 6\right)\cdot 41 + \left(30 a^{2} + 16 a + 12\right)\cdot 41^{2} + \left(2 a^{2} + 18 a + 12\right)\cdot 41^{3} + \left(36 a^{2} + 30 a + 9\right)\cdot 41^{4} + \left(15 a^{2} + 12 a + 39\right)\cdot 41^{5} + \left(27 a^{2} + 38 a + 40\right)\cdot 41^{6} + \left(14 a^{2} + 39 a + 30\right)\cdot 41^{7} + \left(27 a^{2} + 25 a + 25\right)\cdot 41^{8} + \left(26 a^{2} + 12 a + 8\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a^{2} + 32 a + 15 + \left(29 a^{2} + 27 a + 24\right)\cdot 41 + \left(9 a^{2} + 34 a + 22\right)\cdot 41^{2} + \left(24 a + 24\right)\cdot 41^{3} + \left(18 a + 18\right)\cdot 41^{4} + \left(27 a^{2} + 17 a + 27\right)\cdot 41^{5} + \left(21 a^{2} + 19 a + 23\right)\cdot 41^{6} + \left(40 a^{2} + 19 a + 2\right)\cdot 41^{7} + \left(16 a^{2} + 39 a + 13\right)\cdot 41^{8} + \left(20 a^{2} + 6 a + 36\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 + 27\cdot 41 + 33\cdot 41^{2} + 8\cdot 41^{3} + 26\cdot 41^{4} + 12\cdot 41^{5} + 13\cdot 41^{6} + 32\cdot 41^{7} + 35\cdot 41^{8} + 13\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 3 a^{2} + 6 a + 12 + \left(21 a^{2} + 31 a + 32\right)\cdot 41 + \left(2 a^{2} + 36 a + 17\right)\cdot 41^{2} + \left(19 a^{2} + 29 a + 23\right)\cdot 41^{3} + \left(8 a^{2} + 15 a + 10\right)\cdot 41^{4} + \left(39 a^{2} + 9 a + 8\right)\cdot 41^{5} + \left(24 a^{2} + 31 a + 12\right)\cdot 41^{6} + \left(26 a^{2} + 24 a + 34\right)\cdot 41^{7} + \left(33 a^{2} + 30 a + 37\right)\cdot 41^{8} + \left(13 a^{2} + 31 a + 31\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 23 + 25\cdot 41 + 24\cdot 41^{2} + 9\cdot 41^{3} + 3\cdot 41^{4} + 37\cdot 41^{5} + 13\cdot 41^{6} + 18\cdot 41^{7} + 18\cdot 41^{8} + 27\cdot 41^{9} +O(41^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(3,6)(5,7)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $2$ |
| $12$ | $2$ | $(3,5)$ | $0$ |
| $24$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,2)(3,5)$ | $-2$ |
| $36$ | $2$ | $(1,4)(2,8)(3,5)$ | $0$ |
| $16$ | $3$ | $(5,7,6)$ | $0$ |
| $64$ | $3$ | $(2,8,4)(5,7,6)$ | $0$ |
| $12$ | $4$ | $(3,5,6,7)$ | $0$ |
| $36$ | $4$ | $(1,2,4,8)(3,5,6,7)$ | $-2$ |
| $36$ | $4$ | $(1,2,4,8)(3,6)(5,7)$ | $0$ |
| $72$ | $4$ | $(1,3,4,6)(2,5,8,7)$ | $0$ |
| $72$ | $4$ | $(1,2,4,8)(3,5)$ | $2$ |
| $144$ | $4$ | $(1,3,2,5)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,4)(2,8)(5,6,7)$ | $0$ |
| $96$ | $6$ | $(2,8,4)(3,5)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,8,7,4,6)$ | $0$ |
| $144$ | $8$ | $(1,3,2,5,4,6,8,7)$ | $0$ |
| $96$ | $12$ | $(1,2,4,8)(5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.