Basic invariants
| Dimension: | $8$ |
| Group: | $((C_3^2:Q_8):C_3):C_2$ |
| Conductor: | \(45537538411\)\(\medspace = 3571^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.3.45537538411.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $((C_3^2:Q_8):C_3):C_2$ |
| Parity: | odd |
| Projective image: | $C_3^2:\GL(2,3)$ |
| Projective field: | Galois closure of 9.3.45537538411.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{4} + 8x^{2} + 10x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{3} + 2 a^{2} + 8 a + 6 + \left(6 a^{3} + a^{2} + 3 a + 10\right)\cdot 11 + \left(2 a^{3} + a^{2} + 4 a + 7\right)\cdot 11^{2} + \left(9 a^{3} + 7 a^{2} + 10\right)\cdot 11^{3} + \left(3 a^{3} + 9 a^{2} + 10 a + 4\right)\cdot 11^{4} + \left(a^{3} + 9 a^{2} + 4 a + 10\right)\cdot 11^{5} + \left(9 a^{3} + 4 a^{2} + 5 a + 8\right)\cdot 11^{6} + \left(7 a^{3} + 5 a + 8\right)\cdot 11^{7} + \left(6 a^{3} + 4 a^{2} + 6\right)\cdot 11^{8} + \left(2 a^{2} + 6 a + 9\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a^{3} + 6 a^{2} + 5 a + 1 + \left(8 a^{3} + 8 a^{2} + 3 a + 2\right)\cdot 11 + \left(4 a^{3} + 6 a^{2} + 10 a + 8\right)\cdot 11^{2} + \left(5 a^{3} + 5 a^{2} + 2 a + 3\right)\cdot 11^{3} + \left(3 a^{3} + 8 a^{2} + 8 a + 3\right)\cdot 11^{4} + \left(8 a^{3} + 8 a^{2} + 6 a + 3\right)\cdot 11^{5} + \left(9 a^{3} + 6 a^{2} + 10 a + 10\right)\cdot 11^{6} + \left(5 a^{3} + 8 a^{2} + 9 a + 4\right)\cdot 11^{7} + \left(9 a^{3} + 4 a^{2} + a + 3\right)\cdot 11^{8} + \left(2 a^{3} + 9 a^{2} + 5\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a^{2} + 2 a + 9 + \left(3 a^{3} + 6 a^{2} + 7 a + 8\right)\cdot 11 + \left(7 a^{3} + 10 a^{2} + 3\right)\cdot 11^{2} + \left(7 a^{3} + 9 a + 7\right)\cdot 11^{3} + \left(10 a^{3} + 6 a^{2} + 9 a + 3\right)\cdot 11^{4} + \left(3 a^{3} + 2 a^{2} + 8 a + 6\right)\cdot 11^{5} + \left(4 a^{3} + 7 a^{2} + 10 a + 4\right)\cdot 11^{6} + \left(a^{2} + 2 a + 7\right)\cdot 11^{7} + \left(6 a^{3} + 10 a^{2} + 7 a + 9\right)\cdot 11^{8} + \left(9 a^{3} + a^{2} + 8 a + 8\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a^{3} + 3 a^{2} + 7 a + 2 + \left(10 a^{3} + 5 a^{2} + 2 a\right)\cdot 11 + \left(10 a^{2} + 10 a + 10\right)\cdot 11^{2} + \left(10 a^{3} + 8 a^{2} + 5 a + 2\right)\cdot 11^{3} + \left(10 a^{3} + 8 a^{2} + 8 a + 1\right)\cdot 11^{4} + \left(4 a^{3} + 6 a^{2} + 8 a + 7\right)\cdot 11^{5} + \left(2 a^{3} + 2 a^{2} + 10 a + 4\right)\cdot 11^{6} + \left(6 a^{3} + 10 a^{2} + 4 a + 8\right)\cdot 11^{7} + \left(5 a^{3} + 7 a^{2} + 8 a + 5\right)\cdot 11^{8} + \left(7 a^{3} + 10 a^{2} + a + 3\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{3} + 2 a^{2} + 2 a + 2 + \left(8 a^{3} + a^{2} + 4 a + 3\right)\cdot 11 + \left(10 a^{3} + 2 a^{2} + 6\right)\cdot 11^{2} + \left(3 a^{2} + 10 a + 10\right)\cdot 11^{3} + \left(8 a^{3} + 8 a^{2} + 4 a + 10\right)\cdot 11^{4} + \left(4 a^{3} + 8 a^{2} + 1\right)\cdot 11^{5} + \left(4 a^{3} + 5 a^{2} + 9 a + 10\right)\cdot 11^{6} + \left(3 a^{3} + 6 a^{2} + 4 a + 10\right)\cdot 11^{7} + \left(9 a^{3} + 7 a^{2} + 7 a + 4\right)\cdot 11^{8} + \left(2 a^{3} + a^{2} + 7 a + 9\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 7 a + \left(4 a^{3} + 5 a^{2} + 7 a + 9\right)\cdot 11 + \left(7 a^{3} + 3 a^{2} + 6 a + 3\right)\cdot 11^{2} + \left(10 a^{3} + 8 a^{2} + 9 a + 4\right)\cdot 11^{3} + \left(3 a^{3} + 8 a^{2} + 4 a + 2\right)\cdot 11^{4} + \left(8 a^{3} + a + 10\right)\cdot 11^{5} + \left(9 a^{3} + 3 a^{2} + 6 a\right)\cdot 11^{6} + \left(7 a^{3} + 3 a + 3\right)\cdot 11^{7} + \left(10 a^{3} + 3 a^{2} + a + 5\right)\cdot 11^{8} + \left(8 a^{3} + 8 a^{2} + 7 a + 2\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 3 + 11 + 11^{2} + 11^{3} + 9\cdot 11^{4} + 11^{5} + 6\cdot 11^{6} + 6\cdot 11^{7} + 3\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 a^{3} + 10 a^{2} + 4 a + 10 + \left(9 a^{3} + 7 a^{2} + 5 a + 3\right)\cdot 11 + \left(8 a^{3} + 7 a^{2} + 7 a + 3\right)\cdot 11^{2} + \left(2 a^{3} + 3 a^{2} + 10 a + 4\right)\cdot 11^{3} + \left(8 a^{3} + 4 a^{2} + 10 a + 7\right)\cdot 11^{4} + \left(4 a^{3} + 6 a^{2} + 8 a + 3\right)\cdot 11^{5} + \left(3 a^{3} + 4 a^{2} + 7 a + 3\right)\cdot 11^{6} + \left(a^{3} + a^{2} + 2\right)\cdot 11^{7} + \left(7 a^{3} + 9 a^{2} + 2 a\right)\cdot 11^{8} + \left(10 a^{3} + 6 a^{2} + 9 a + 6\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 6 a^{3} + 7 a^{2} + 9 a + 3 + \left(3 a^{3} + 7 a^{2} + 9 a + 5\right)\cdot 11 + \left(a^{3} + a^{2} + 3 a + 10\right)\cdot 11^{2} + \left(8 a^{3} + 6 a^{2} + 6 a + 9\right)\cdot 11^{3} + \left(5 a^{3} + 8 a\right)\cdot 11^{4} + \left(7 a^{3} + 3 a + 10\right)\cdot 11^{5} + \left(9 a^{2} + 5 a + 5\right)\cdot 11^{6} + \left(3 a^{2} + 2\right)\cdot 11^{7} + \left(8 a^{2} + 4 a + 4\right)\cdot 11^{8} + \left(a^{3} + 2 a^{2} + 3 a + 5\right)\cdot 11^{9} +O(11^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,7)(2,4)(3,9)$ | $2$ |
| $8$ | $3$ | $(1,2,5)(3,6,4)(7,8,9)$ | $-1$ |
| $24$ | $3$ | $(1,8,3)(2,6,9)$ | $2$ |
| $48$ | $3$ | $(1,9,5)(2,4,3)(6,7,8)$ | $-1$ |
| $54$ | $4$ | $(1,5,4,3)(2,7,6,8)$ | $0$ |
| $72$ | $6$ | $(1,2,9,7,4,3)(5,6,8)$ | $-1$ |
| $72$ | $6$ | $(1,2,6,8,7,4)(5,9)$ | $0$ |
| $54$ | $8$ | $(1,7,5,6,4,8,3,2)$ | $0$ |
| $54$ | $8$ | $(1,8,5,2,4,7,3,6)$ | $0$ |