Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(519718464000\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 67^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.65289632040.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | even |
| Determinant: | 1.8040.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.65289632040.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 3x^{6} + 11x^{5} - 5x^{4} - 12x^{3} - 34x^{2} - 7x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 42\cdot 73 + 9\cdot 73^{2} + 51\cdot 73^{3} + 27\cdot 73^{4} + 72\cdot 73^{5} + 39\cdot 73^{6} + 69\cdot 73^{7} + 19\cdot 73^{8} + 22\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 a + 33 + \left(60 a + 61\right)\cdot 73 + \left(24 a + 5\right)\cdot 73^{2} + \left(47 a + 15\right)\cdot 73^{3} + \left(71 a + 29\right)\cdot 73^{4} + \left(30 a + 39\right)\cdot 73^{5} + \left(59 a + 14\right)\cdot 73^{6} + \left(53 a + 38\right)\cdot 73^{7} + \left(50 a + 58\right)\cdot 73^{8} + \left(10 a + 38\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 40 + \left(12 a + 22\right)\cdot 73 + \left(48 a + 45\right)\cdot 73^{2} + \left(33 a + 61\right)\cdot 73^{3} + \left(5 a + 47\right)\cdot 73^{4} + \left(61 a + 54\right)\cdot 73^{5} + \left(51 a + 16\right)\cdot 73^{6} + \left(40 a + 63\right)\cdot 73^{7} + \left(44 a + 33\right)\cdot 73^{8} + \left(30 a + 30\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a + 68 + \left(62 a + 48\right)\cdot 73 + \left(20 a + 23\right)\cdot 73^{2} + \left(44 a + 16\right)\cdot 73^{3} + \left(40 a + 67\right)\cdot 73^{4} + \left(38 a + 21\right)\cdot 73^{5} + \left(38 a + 19\right)\cdot 73^{6} + \left(51 a + 35\right)\cdot 73^{7} + \left(72 a + 64\right)\cdot 73^{8} + \left(68 a + 12\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 a + 65 + \left(12 a + 61\right)\cdot 73 + \left(48 a + 19\right)\cdot 73^{2} + \left(25 a + 59\right)\cdot 73^{3} + \left(a + 50\right)\cdot 73^{4} + \left(42 a + 60\right)\cdot 73^{5} + \left(13 a + 15\right)\cdot 73^{6} + \left(19 a + 67\right)\cdot 73^{7} + \left(22 a + 10\right)\cdot 73^{8} + \left(62 a + 20\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 a + 27 + \left(60 a + 39\right)\cdot 73 + \left(24 a + 31\right)\cdot 73^{2} + \left(39 a + 41\right)\cdot 73^{3} + \left(67 a + 30\right)\cdot 73^{4} + \left(11 a + 13\right)\cdot 73^{5} + \left(21 a + 38\right)\cdot 73^{6} + \left(32 a + 60\right)\cdot 73^{7} + \left(28 a + 53\right)\cdot 73^{8} + \left(42 a + 4\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 29 a + 54 + \left(10 a + 46\right)\cdot 73 + \left(52 a + 23\right)\cdot 73^{2} + \left(28 a + 55\right)\cdot 73^{3} + \left(32 a + 71\right)\cdot 73^{4} + \left(34 a + 23\right)\cdot 73^{5} + \left(34 a + 23\right)\cdot 73^{6} + \left(21 a + 5\right)\cdot 73^{7} + 12\cdot 73^{8} + \left(4 a + 1\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 65 + 41\cdot 73 + 59\cdot 73^{2} + 64\cdot 73^{3} + 39\cdot 73^{4} + 5\cdot 73^{5} + 51\cdot 73^{6} + 25\cdot 73^{7} + 38\cdot 73^{8} + 15\cdot 73^{9} +O(73^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,5)(4,7)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $1$ |
| $12$ | $2$ | $(1,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $1$ |
| $36$ | $2$ | $(1,3)(2,5)(4,7)$ | $-1$ |
| $16$ | $3$ | $(1,6,8)$ | $0$ |
| $64$ | $3$ | $(1,6,8)(4,5,7)$ | $0$ |
| $12$ | $4$ | $(2,4,5,7)$ | $-3$ |
| $36$ | $4$ | $(1,3,6,8)(2,4,5,7)$ | $1$ |
| $36$ | $4$ | $(1,3,6,8)(2,5)(4,7)$ | $1$ |
| $72$ | $4$ | $(1,2,6,5)(3,4,8,7)$ | $-1$ |
| $72$ | $4$ | $(1,3)(2,4,5,7)$ | $-1$ |
| $144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,8,6)(2,5)(4,7)$ | $0$ |
| $96$ | $6$ | $(1,3)(4,7,5)$ | $0$ |
| $192$ | $6$ | $(1,4,6,5,8,7)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,4,6,5,8,7)$ | $-1$ |
| $96$ | $12$ | $(1,6,8)(2,4,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.