Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(540\!\cdots\!376\)\(\medspace = 2^{12} \cdot 3^{22} \cdot 19^{13} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.411557366988.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.411557366988.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{6} - 19x^{5} - 18x^{4} + 28x^{2} - 48 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{3} + 2x + 68 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a^{2} + 29 a + 21 + \left(11 a^{2} + 57 a + 28\right)\cdot 73 + \left(33 a^{2} + 13 a + 18\right)\cdot 73^{2} + \left(42 a^{2} + 29 a + 41\right)\cdot 73^{3} + \left(71 a^{2} + 59 a + 8\right)\cdot 73^{4} + \left(27 a^{2} + 65\right)\cdot 73^{5} + \left(8 a^{2} + 48 a + 40\right)\cdot 73^{6} + \left(61 a^{2} + 3 a + 61\right)\cdot 73^{7} + \left(5 a^{2} + 39 a + 71\right)\cdot 73^{8} + \left(6 a^{2} + 58 a + 27\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 11\cdot 73 + 58\cdot 73^{2} + 58\cdot 73^{3} + 41\cdot 73^{4} + 47\cdot 73^{5} + 2\cdot 73^{6} + 16\cdot 73^{7} + 65\cdot 73^{8} + 30\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 67 a^{2} + 57 a + 5 + \left(21 a^{2} + 31 a + 67\right)\cdot 73 + \left(60 a^{2} + 49 a + 5\right)\cdot 73^{2} + \left(42 a^{2} + 23 a + 66\right)\cdot 73^{3} + \left(45 a^{2} + 47 a + 46\right)\cdot 73^{4} + \left(19 a^{2} + 41 a + 29\right)\cdot 73^{5} + \left(19 a^{2} + 25 a + 55\right)\cdot 73^{6} + \left(35 a^{2} + 41 a + 2\right)\cdot 73^{7} + \left(49 a^{2} + 58 a + 57\right)\cdot 73^{8} + \left(63 a^{2} + 48 a + 31\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 32\cdot 73 + 4\cdot 73^{2} + 46\cdot 73^{3} + 41\cdot 73^{4} + 62\cdot 73^{5} + 56\cdot 73^{6} + 59\cdot 73^{7} + 26\cdot 73^{8} + 13\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a^{2} + 22 a + 20 + \left(58 a^{2} + 49 a + 25\right)\cdot 73 + \left(19 a^{2} + 2 a + 31\right)\cdot 73^{2} + \left(43 a^{2} + 51 a + 62\right)\cdot 73^{3} + \left(17 a^{2} + 22 a + 33\right)\cdot 73^{4} + \left(51 a^{2} + 49 a + 52\right)\cdot 73^{5} + \left(50 a^{2} + a + 66\right)\cdot 73^{6} + \left(33 a^{2} + 46 a + 63\right)\cdot 73^{7} + \left(5 a^{2} + 65 a + 9\right)\cdot 73^{8} + \left(19 a^{2} + 42 a + 15\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 60 a + 13 + \left(40 a^{2} + 56 a + 18\right)\cdot 73 + \left(52 a^{2} + 9 a + 44\right)\cdot 73^{2} + \left(60 a^{2} + 20 a + 65\right)\cdot 73^{3} + \left(28 a^{2} + 39 a + 48\right)\cdot 73^{4} + \left(25 a^{2} + 30 a + 61\right)\cdot 73^{5} + \left(45 a^{2} + 72 a + 65\right)\cdot 73^{6} + \left(49 a^{2} + 27 a + 21\right)\cdot 73^{7} + \left(17 a^{2} + 48 a + 63\right)\cdot 73^{8} + \left(3 a^{2} + 38 a + 72\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 15 a^{2} + 5 a + 31 + \left(30 a^{2} + 35 a + 36\right)\cdot 73 + \left(69 a^{2} + 10 a + 24\right)\cdot 73^{2} + \left(37 a^{2} + 2 a + 55\right)\cdot 73^{3} + \left(38 a^{2} + 33 a + 61\right)\cdot 73^{4} + \left(9 a^{2} + 64 a + 69\right)\cdot 73^{5} + \left(63 a^{2} + 14 a + 58\right)\cdot 73^{6} + \left(42 a^{2} + 69 a + 51\right)\cdot 73^{7} + \left(a^{2} + 58 a + 4\right)\cdot 73^{8} + \left(59 a^{2} + 61 a + 44\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 33 a^{2} + 46 a + 55 + \left(57 a^{2} + 61 a + 72\right)\cdot 73 + \left(56 a^{2} + 59 a + 31\right)\cdot 73^{2} + \left(64 a^{2} + 19 a + 42\right)\cdot 73^{3} + \left(16 a^{2} + 17 a + 8\right)\cdot 73^{4} + \left(12 a^{2} + 32 a + 49\right)\cdot 73^{5} + \left(32 a^{2} + 56 a + 17\right)\cdot 73^{6} + \left(69 a^{2} + 30 a + 14\right)\cdot 73^{7} + \left(65 a^{2} + 21 a + 66\right)\cdot 73^{8} + \left(67 a^{2} + 41 a + 55\right)\cdot 73^{9} +O(73^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,4)(3,6)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,7)(5,8)$ | $0$ |
| $16$ | $3$ | $(3,4,6)$ | $0$ |
| $64$ | $3$ | $(3,4,6)(5,7,8)$ | $0$ |
| $12$ | $4$ | $(1,3,4,6)$ | $0$ |
| $36$ | $4$ | $(1,3,4,6)(2,5,7,8)$ | $-2$ |
| $36$ | $4$ | $(1,4)(2,5,7,8)(3,6)$ | $0$ |
| $72$ | $4$ | $(1,7,4,2)(3,8,6,5)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,5,7,8)$ | $2$ |
| $144$ | $4$ | $(1,5,3,2)(4,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(2,7)(3,6,4)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(5,8,7)$ | $0$ |
| $192$ | $6$ | $(1,2)(3,7,4,8,6,5)$ | $0$ |
| $144$ | $8$ | $(1,5,3,7,4,8,6,2)$ | $0$ |
| $96$ | $12$ | $(2,5,7,8)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.