Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(586\!\cdots\!208\)\(\medspace = 2^{61} \cdot 3^{26} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.3057647616.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.3057647616.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 12x^{5} + 9x^{4} - 8x^{2} + 8x - 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 11 a + 2 + \left(12 a^{2} + 10 a + 14\right)\cdot 17 + \left(9 a^{2} + a + 5\right)\cdot 17^{2} + \left(a^{2} + 9 a + 3\right)\cdot 17^{3} + \left(11 a^{2} + 16 a + 16\right)\cdot 17^{4} + \left(7 a^{2} + 2 a + 11\right)\cdot 17^{5} + \left(5 a^{2} + 5 a + 6\right)\cdot 17^{6} + \left(7 a^{2} + 14 a + 9\right)\cdot 17^{7} + \left(11 a^{2} + 13 a + 10\right)\cdot 17^{8} + \left(7 a^{2} + 15 a + 9\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 13 + \left(12 a^{2} + 12 a\right)\cdot 17 + \left(8 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(8 a^{2} + 15 a + 5\right)\cdot 17^{3} + \left(12 a^{2} + 11 a + 1\right)\cdot 17^{4} + \left(2 a^{2} + 3 a + 12\right)\cdot 17^{5} + \left(3 a^{2} + 2\right)\cdot 17^{6} + \left(15 a^{2} + 4 a + 8\right)\cdot 17^{7} + \left(5 a^{2} + 4 a + 9\right)\cdot 17^{8} + \left(7 a^{2} + 7 a + 13\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a^{2} + 3 a + 7 + \left(3 a^{2} + 5 a + 6\right)\cdot 17 + \left(16 a^{2} + a + 15\right)\cdot 17^{2} + \left(5 a^{2} + 14 a + 9\right)\cdot 17^{3} + \left(12 a^{2} + 15 a + 12\right)\cdot 17^{4} + 2 a^{2} 17^{5} + 16 a^{2} 17^{6} + \left(10 a^{2} + 6 a + 11\right)\cdot 17^{7} + \left(5 a^{2} + 13 a + 3\right)\cdot 17^{8} + \left(4 a^{2} + 3 a\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 16 a + 2 + \left(a^{2} + 15 a + 5\right)\cdot 17 + \left(9 a^{2} + 3 a + 16\right)\cdot 17^{2} + \left(2 a^{2} + 4 a + 1\right)\cdot 17^{3} + \left(9 a^{2} + 6 a + 16\right)\cdot 17^{4} + \left(11 a^{2} + 12 a\right)\cdot 17^{5} + \left(14 a^{2} + 16 a + 16\right)\cdot 17^{6} + \left(7 a^{2} + 6 a + 8\right)\cdot 17^{7} + \left(5 a^{2} + 16 a + 3\right)\cdot 17^{8} + \left(5 a^{2} + 5 a + 12\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 4\cdot 17 + 3\cdot 17^{2} + 16\cdot 17^{3} + 3\cdot 17^{4} + 3\cdot 17^{5} + 15\cdot 17^{6} + 5\cdot 17^{7} + 8\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a^{2} + 4 a + 4 + \left(12 a + 6\right)\cdot 17 + \left(7 a^{2} + 2 a + 15\right)\cdot 17^{2} + \left(5 a^{2} + 6 a + 5\right)\cdot 17^{3} + \left(9 a^{2} + 9 a + 9\right)\cdot 17^{4} + \left(8 a^{2} + 11 a + 12\right)\cdot 17^{5} + \left(2 a + 14\right)\cdot 17^{6} + \left(11 a^{2} + 14 a + 11\right)\cdot 17^{7} + \left(2 a^{2} + 5 a + 4\right)\cdot 17^{8} + \left(7 a^{2} + 2 a + 9\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 + 16\cdot 17 + 17^{2} + 10\cdot 17^{3} + 7\cdot 17^{4} + 13\cdot 17^{5} + 7\cdot 17^{6} + 3\cdot 17^{7} + 8\cdot 17^{8} + 3\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 15 a^{2} + 2 a + 1 + \left(3 a^{2} + 11 a + 14\right)\cdot 17 + \left(12 a + 10\right)\cdot 17^{2} + \left(10 a^{2} + a + 14\right)\cdot 17^{3} + \left(13 a^{2} + 8 a\right)\cdot 17^{4} + \left(2 a + 13\right)\cdot 17^{5} + \left(11 a^{2} + 9 a + 4\right)\cdot 17^{6} + \left(15 a^{2} + 5 a + 9\right)\cdot 17^{7} + \left(2 a^{2} + 14 a + 10\right)\cdot 17^{8} + \left(2 a^{2} + 15 a + 11\right)\cdot 17^{9} +O(17^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,4)(3,5)$ | $-6$ |
| $9$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $2$ |
| $12$ | $2$ | $(1,6)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $36$ | $2$ | $(1,6)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ |
| $16$ | $3$ | $(1,7,8)$ | $0$ |
| $64$ | $3$ | $(1,7,8)(3,4,5)$ | $0$ |
| $12$ | $4$ | $(2,3,4,5)$ | $0$ |
| $36$ | $4$ | $(1,6,7,8)(2,3,4,5)$ | $-2$ |
| $36$ | $4$ | $(1,6,7,8)(2,4)(3,5)$ | $0$ |
| $72$ | $4$ | $(1,2,7,4)(3,8,5,6)$ | $0$ |
| $72$ | $4$ | $(1,6)(2,3,4,5)$ | $2$ |
| $144$ | $4$ | $(1,3,6,2)(4,7)(5,8)$ | $0$ |
| $48$ | $6$ | $(1,8,7)(2,4)(3,5)$ | $0$ |
| $96$ | $6$ | $(1,6)(3,5,4)$ | $0$ |
| $192$ | $6$ | $(1,3,7,4,8,5)(2,6)$ | $0$ |
| $144$ | $8$ | $(1,2,6,3,7,4,8,5)$ | $0$ |
| $96$ | $12$ | $(1,7,8)(2,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.