Basic invariants
| Dimension: | $12$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(59006820995623747584\)\(\medspace = 2^{16} \cdot 3^{15} \cdot 13^{7} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.831506217984.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1763 |
| Parity: | odd |
| Determinant: | 1.39.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.831506217984.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 10x^{6} - 28x^{5} + 44x^{4} - 22x^{3} - 20x^{2} + 28x - 23 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 6 a + 9 + \left(a^{2} + 3 a + 7\right)\cdot 11 + \left(7 a^{2} + 8\right)\cdot 11^{2} + \left(a^{2} + 4 a + 9\right)\cdot 11^{3} + \left(2 a^{2} + 4 a + 7\right)\cdot 11^{4} + \left(4 a^{2} + 3 a + 6\right)\cdot 11^{5} + \left(9 a^{2} + a + 5\right)\cdot 11^{6} + \left(5 a^{2} + 2 a + 9\right)\cdot 11^{7} + \left(2 a^{2} + 1\right)\cdot 11^{8} + \left(3 a^{2} + a + 2\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 5\cdot 11 + 3\cdot 11^{3} + 10\cdot 11^{4} + 11^{5} + 2\cdot 11^{6} + 5\cdot 11^{7} + 4\cdot 11^{8} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a^{2} + 2 a + 1 + \left(2 a^{2} + 6 a + 6\right)\cdot 11 + \left(a^{2} + a + 4\right)\cdot 11^{2} + \left(a^{2} + 2 a + 5\right)\cdot 11^{3} + \left(3 a^{2} + a + 5\right)\cdot 11^{4} + \left(5 a^{2} + 7 a + 4\right)\cdot 11^{5} + \left(2 a^{2} + 8 a + 7\right)\cdot 11^{6} + \left(3 a^{2} + 2 a + 9\right)\cdot 11^{7} + \left(a^{2} + 9 a + 3\right)\cdot 11^{8} + \left(a^{2} + a + 10\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 + 5\cdot 11 + 8\cdot 11^{2} + 7\cdot 11^{3} + 8\cdot 11^{4} + 5\cdot 11^{5} + 8\cdot 11^{6} + 10\cdot 11^{7} + 7\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a^{2} + a + 4 + \left(9 a^{2} + 6\right)\cdot 11 + \left(9 a^{2} + 3 a + 3\right)\cdot 11^{2} + \left(4 a^{2} + 8 a + 6\right)\cdot 11^{3} + \left(7 a^{2} + 5 a + 9\right)\cdot 11^{4} + \left(5 a^{2} + 4 a + 3\right)\cdot 11^{5} + \left(2 a^{2} + 4 a + 10\right)\cdot 11^{6} + \left(8 a^{2} + 10 a + 3\right)\cdot 11^{7} + \left(7 a + 9\right)\cdot 11^{8} + \left(7 a^{2} + 10 a + 9\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a^{2} + 8 a + \left(a^{2} + 2 a + 10\right)\cdot 11 + \left(4 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + 8 a + 2\right)\cdot 11^{3} + \left(9 a^{2} + 6 a + 1\right)\cdot 11^{4} + \left(7 a^{2} + 8 a + 3\right)\cdot 11^{5} + \left(8 a^{2} + 10 a\right)\cdot 11^{6} + \left(2 a^{2} + 7 a + 4\right)\cdot 11^{7} + \left(3 a + 1\right)\cdot 11^{8} + \left(6 a^{2} + 10 a + 1\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 4 a^{2} + 2 a + 10 + \left(8 a + 8\right)\cdot 11 + \left(a^{2} + 3 a + 2\right)\cdot 11^{2} + \left(a^{2} + 5 a + 1\right)\cdot 11^{3} + \left(5 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(8 a^{2} + 8 a + 3\right)\cdot 11^{5} + \left(10 a^{2} + 6 a + 10\right)\cdot 11^{6} + \left(10 a^{2} + 3 a + 3\right)\cdot 11^{7} + \left(9 a^{2} + 10 a + 3\right)\cdot 11^{8} + \left(8 a^{2} + 1\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 4 a^{2} + 3 a + 2 + \left(7 a^{2} + a + 5\right)\cdot 11 + \left(2 a^{2} + 9 a + 6\right)\cdot 11^{2} + \left(8 a^{2} + 4 a + 7\right)\cdot 11^{3} + \left(5 a^{2} + 5 a + 1\right)\cdot 11^{4} + \left(a^{2} + 3\right)\cdot 11^{5} + \left(10 a^{2} + a + 10\right)\cdot 11^{6} + \left(a^{2} + 6 a + 7\right)\cdot 11^{7} + \left(7 a^{2} + a\right)\cdot 11^{8} + \left(6 a^{2} + 8 a + 3\right)\cdot 11^{9} +O(11^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $6$ | $2$ | $(1,4)(3,8)$ | $4$ |
| $9$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-4$ |
| $12$ | $2$ | $(1,3)$ | $-2$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,6)(5,7)$ | $-2$ |
| $16$ | $3$ | $(3,4,8)$ | $-3$ |
| $64$ | $3$ | $(3,4,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(1,3,4,8)$ | $2$ |
| $36$ | $4$ | $(1,3,4,8)(2,5,6,7)$ | $0$ |
| $36$ | $4$ | $(1,4)(2,5,6,7)(3,8)$ | $2$ |
| $72$ | $4$ | $(1,6,4,2)(3,7,8,5)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,5,6,7)$ | $0$ |
| $144$ | $4$ | $(1,5,3,2)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(2,6)(3,8,4)(5,7)$ | $1$ |
| $96$ | $6$ | $(1,3)(2,6,7)$ | $1$ |
| $192$ | $6$ | $(1,2)(3,6,4,7,8,5)$ | $0$ |
| $144$ | $8$ | $(1,5,3,6,4,7,8,2)$ | $0$ |
| $96$ | $12$ | $(2,5,6,7)(3,4,8)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.