Basic invariants
Dimension: | $18$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(376\!\cdots\!000\)\(\medspace = 2^{18} \cdot 5^{12} \cdot 73^{9} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.1135929640000.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 36T1758 |
Parity: | odd |
Determinant: | 1.292.2t1.a.a |
Projective image: | $S_4\wr C_2$ |
Projective stem field: | Galois closure of 8.2.1135929640000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} - 4x^{6} - 13x^{5} - 6x^{4} - 13x^{3} + 41x^{2} + 28x + 38 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{3} + 3x + 86 \)
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 80\cdot 89 + 77\cdot 89^{2} + 47\cdot 89^{3} + 56\cdot 89^{4} + 60\cdot 89^{5} + 13\cdot 89^{6} + 89^{7} + 31\cdot 89^{8} + 43\cdot 89^{9} +O(89^{10})\) |
$r_{ 2 }$ | $=$ | \( 6 + 19\cdot 89 + 59\cdot 89^{2} + 31\cdot 89^{3} + 26\cdot 89^{4} + 32\cdot 89^{5} + 71\cdot 89^{6} + 67\cdot 89^{7} + 89^{8} + 31\cdot 89^{9} +O(89^{10})\) |
$r_{ 3 }$ | $=$ | \( 54 a^{2} + 11 a + 26 + \left(48 a^{2} + 48 a + 28\right)\cdot 89 + \left(a^{2} + 45 a + 36\right)\cdot 89^{2} + \left(66 a^{2} + 58 a + 72\right)\cdot 89^{3} + \left(7 a^{2} + 7 a + 18\right)\cdot 89^{4} + \left(3 a^{2} + 39 a + 74\right)\cdot 89^{5} + \left(30 a^{2} + 45 a + 46\right)\cdot 89^{6} + \left(35 a^{2} + 81 a + 69\right)\cdot 89^{7} + \left(27 a^{2} + 6 a + 8\right)\cdot 89^{8} + \left(54 a^{2} + 25 a + 24\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 4 }$ | $=$ | \( 71 a^{2} + 56 a + 60 + \left(65 a^{2} + 74 a + 62\right)\cdot 89 + \left(28 a^{2} + 44 a + 1\right)\cdot 89^{2} + \left(21 a^{2} + 68 a + 72\right)\cdot 89^{3} + \left(17 a^{2} + 37 a + 37\right)\cdot 89^{4} + \left(27 a^{2} + 40 a + 33\right)\cdot 89^{5} + \left(29 a^{2} + 78 a + 45\right)\cdot 89^{6} + \left(69 a^{2} + 6 a + 48\right)\cdot 89^{7} + \left(63 a^{2} + 16 a + 81\right)\cdot 89^{8} + \left(44 a^{2} + 69 a + 4\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 5 }$ | $=$ | \( 53 a^{2} + 22 a + 24 + \left(63 a^{2} + 55 a + 58\right)\cdot 89 + \left(58 a^{2} + 87 a + 61\right)\cdot 89^{2} + \left(a^{2} + 50 a + 32\right)\cdot 89^{3} + \left(64 a^{2} + 43 a + 42\right)\cdot 89^{4} + \left(58 a^{2} + 9 a + 7\right)\cdot 89^{5} + \left(29 a^{2} + 54 a + 46\right)\cdot 89^{6} + \left(73 a^{2} + 56\right)\cdot 89^{7} + \left(86 a^{2} + 66 a + 38\right)\cdot 89^{8} + \left(78 a^{2} + 83 a + 73\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 6 }$ | $=$ | \( 84 a^{2} + 33 a + 10 + \left(68 a + 8\right)\cdot 89 + \left(18 a^{2} + 86 a + 46\right)\cdot 89^{2} + \left(47 a^{2} + 19 a + 38\right)\cdot 89^{3} + \left(16 a^{2} + 37 a + 61\right)\cdot 89^{4} + \left(57 a^{2} + 57 a + 74\right)\cdot 89^{5} + \left(18 a^{2} + 69 a + 51\right)\cdot 89^{6} + \left(17 a^{2} + 22 a + 12\right)\cdot 89^{7} + \left(34 a^{2} + 59 a + 44\right)\cdot 89^{8} + \left(69 a^{2} + 51 a + 20\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 7 }$ | $=$ | \( 77 a^{2} + 28 a + 85 + \left(5 a^{2} + 25 a + 17\right)\cdot 89 + \left(43 a^{2} + 25 a + 7\right)\cdot 89^{2} + \left(37 a^{2} + 4 a + 19\right)\cdot 89^{3} + \left(37 a^{2} + 77 a + 14\right)\cdot 89^{4} + \left(72 a^{2} + 88 a + 16\right)\cdot 89^{5} + \left(75 a^{2} + 82 a + 77\right)\cdot 89^{6} + \left(55 a^{2} + 46 a\right)\cdot 89^{7} + \left(a^{2} + 73 a + 68\right)\cdot 89^{8} + \left(54 a^{2} + 47 a + 78\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 8 }$ | $=$ | \( 17 a^{2} + 28 a + 54 + \left(82 a^{2} + 84 a + 81\right)\cdot 89 + \left(27 a^{2} + 65 a + 65\right)\cdot 89^{2} + \left(4 a^{2} + 64 a + 41\right)\cdot 89^{3} + \left(35 a^{2} + 63 a + 9\right)\cdot 89^{4} + \left(48 a^{2} + 31 a + 57\right)\cdot 89^{5} + \left(83 a^{2} + 25 a + 3\right)\cdot 89^{6} + \left(15 a^{2} + 19 a + 10\right)\cdot 89^{7} + \left(53 a^{2} + 45 a + 82\right)\cdot 89^{8} + \left(54 a^{2} + 78 a + 79\right)\cdot 89^{9} +O(89^{10})\) |
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 |
$1$ | $1$ | $()$ | $18$ |
$6$ | $2$ | $(1,4)(3,5)$ | $-6$ |
$9$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $2$ |
$12$ | $2$ | $(2,6)$ | $0$ |
$24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
$36$ | $2$ | $(1,3)(2,6)$ | $-2$ |
$36$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
$16$ | $3$ | $(2,7,8)$ | $0$ |
$64$ | $3$ | $(2,7,8)(3,4,5)$ | $0$ |
$12$ | $4$ | $(1,3,4,5)$ | $0$ |
$36$ | $4$ | $(1,3,4,5)(2,6,7,8)$ | $-2$ |
$36$ | $4$ | $(1,4)(2,6,7,8)(3,5)$ | $0$ |
$72$ | $4$ | $(1,7,4,2)(3,8,5,6)$ | $0$ |
$72$ | $4$ | $(1,3,4,5)(2,6)$ | $2$ |
$144$ | $4$ | $(1,2,3,6)(4,7)(5,8)$ | $0$ |
$48$ | $6$ | $(1,4)(2,8,7)(3,5)$ | $0$ |
$96$ | $6$ | $(2,6)(3,5,4)$ | $0$ |
$192$ | $6$ | $(1,6)(2,3,7,4,8,5)$ | $0$ |
$144$ | $8$ | $(1,6,3,7,4,8,5,2)$ | $0$ |
$96$ | $12$ | $(1,3,4,5)(2,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.