Basic invariants
| Dimension: | $6$ |
| Group: | $A_4\wr C_2$ |
| Conductor: | \(151400448\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 37^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.454201344.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $A_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.454201344.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 4x^{6} + 14x^{5} - 4x^{4} - 22x^{3} + 20x^{2} - 6x + 7 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 2 a + 18 + \left(3 a^{2} + 2 a + 13\right)\cdot 19 + \left(18 a^{2} + 3 a + 1\right)\cdot 19^{2} + \left(8 a^{2} + 4 a + 11\right)\cdot 19^{3} + \left(15 a^{2} + 3 a + 17\right)\cdot 19^{4} + \left(6 a^{2} + 14 a + 9\right)\cdot 19^{5} + \left(7 a^{2} + 3 a + 14\right)\cdot 19^{6} + \left(a^{2} + 2 a + 17\right)\cdot 19^{7} + \left(4 a^{2} + 3 a + 4\right)\cdot 19^{8} + \left(8 a + 14\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a^{2} + 17 a + 5 + \left(18 a^{2} + 6 a + 4\right)\cdot 19 + \left(14 a^{2} + 18 a + 12\right)\cdot 19^{2} + \left(3 a^{2} + 9 a + 3\right)\cdot 19^{3} + \left(14 a^{2} + 17 a + 14\right)\cdot 19^{4} + \left(a^{2} + 2\right)\cdot 19^{5} + \left(5 a^{2} + 9 a + 2\right)\cdot 19^{6} + \left(11 a^{2} + 18 a + 6\right)\cdot 19^{7} + \left(17 a^{2} + 17 a + 9\right)\cdot 19^{8} + \left(a^{2} + 5 a + 12\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 8\cdot 19 + 10\cdot 19^{2} + 18\cdot 19^{3} + 11\cdot 19^{4} + 7\cdot 19^{5} + 13\cdot 19^{6} + 6\cdot 19^{7} + 14\cdot 19^{8} + 4\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a^{2} + 9 a + 14 + \left(16 a^{2} + 5 a + 11\right)\cdot 19 + \left(16 a^{2} + a + 15\right)\cdot 19^{2} + \left(16 a^{2} + 17 a + 12\right)\cdot 19^{3} + \left(16 a^{2} + 5 a + 10\right)\cdot 19^{4} + \left(13 a^{2} + 3 a + 2\right)\cdot 19^{5} + \left(11 a^{2} + 12 a + 10\right)\cdot 19^{6} + \left(18 a^{2} + 15 a + 3\right)\cdot 19^{7} + \left(11 a^{2} + 7 a + 7\right)\cdot 19^{8} + \left(18 a^{2} + 10 a + 5\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a^{2} + 6 a + 5 + \left(13 a^{2} + a + 16\right)\cdot 19 + \left(11 a^{2} + 7 a + 1\right)\cdot 19^{2} + \left(16 a^{2} + 5 a + 12\right)\cdot 19^{3} + \left(18 a^{2} + 3 a + 9\right)\cdot 19^{4} + \left(12 a^{2} + 10 a + 6\right)\cdot 19^{5} + \left(2 a^{2} + 5 a + 11\right)\cdot 19^{6} + \left(7 a^{2} + a + 4\right)\cdot 19^{7} + \left(12 a^{2} + a + 8\right)\cdot 19^{8} + \left(3 a^{2} + 3\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 + 10\cdot 19 + 8\cdot 19^{2} + 2\cdot 19^{3} + 10\cdot 19^{4} + 6\cdot 19^{5} + 8\cdot 19^{6} + 13\cdot 19^{7} + 19^{8} + 11\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 14 a^{2} + 4 a + 18 + \left(8 a^{2} + 12 a + 10\right)\cdot 19 + \left(9 a^{2} + 10 a + 8\right)\cdot 19^{2} + \left(4 a^{2} + 15 a + 11\right)\cdot 19^{3} + \left(2 a^{2} + 9 a + 9\right)\cdot 19^{4} + \left(11 a^{2} + 5 a + 1\right)\cdot 19^{5} + \left(4 a^{2} + a + 10\right)\cdot 19^{6} + \left(12 a^{2} + 2 a + 5\right)\cdot 19^{7} + \left(13 a^{2} + 10 a + 5\right)\cdot 19^{8} + \left(15 a^{2} + 8 a + 10\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 10 a^{2} + 4 + \left(16 a^{2} + 10 a + 18\right)\cdot 19 + \left(4 a^{2} + 16 a + 16\right)\cdot 19^{2} + \left(6 a^{2} + 4 a + 3\right)\cdot 19^{3} + \left(8 a^{2} + 17 a + 11\right)\cdot 19^{4} + \left(10 a^{2} + 3 a\right)\cdot 19^{5} + \left(6 a^{2} + 6 a + 6\right)\cdot 19^{6} + \left(6 a^{2} + 17 a + 18\right)\cdot 19^{7} + \left(16 a^{2} + 16 a + 5\right)\cdot 19^{8} + \left(16 a^{2} + 4 a + 14\right)\cdot 19^{9} +O(19^{10})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $6$ | |
| $6$ | $2$ | $(1,3)(2,8)$ | $2$ | |
| $9$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $-2$ | |
| $12$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ | ✓ |
| $8$ | $3$ | $(1,3,8)$ | $3$ | |
| $8$ | $3$ | $(1,8,3)$ | $3$ | |
| $16$ | $3$ | $(1,3,8)(4,5,7)$ | $0$ | |
| $16$ | $3$ | $(1,8,3)(4,7,5)$ | $0$ | |
| $32$ | $3$ | $(1,3,8)(4,6,7)$ | $0$ | |
| $36$ | $4$ | $(1,7,3,5)(2,4,8,6)$ | $0$ | |
| $24$ | $6$ | $(1,3)(2,8)(4,6,7)$ | $-1$ | |
| $24$ | $6$ | $(1,3)(2,8)(4,7,6)$ | $-1$ | |
| $48$ | $6$ | $(1,7,3,4,8,5)(2,6)$ | $0$ | |
| $48$ | $6$ | $(1,5,8,4,3,7)(2,6)$ | $0$ |