Basic invariants
| Dimension: | $6$ |
| Group: | $\PGL(2,7)$ |
| Conductor: | \(7802490888622528\)\(\medspace = 2^{6} \cdot 7^{7} \cdot 23^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.7802490888622528.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T82 |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $\PGL(2,7)$ |
| Projective stem field: | Galois closure of 8.2.7802490888622528.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 21x^{6} + 91x^{5} - 819x^{4} + 2765x^{3} - 9779x^{2} + 22167x - 17946 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 1 + 5\cdot 11 + 10\cdot 11^{2} + 9\cdot 11^{3} + 7\cdot 11^{4} + 8\cdot 11^{5} + 8\cdot 11^{6} + 10\cdot 11^{7} + 11^{8} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a^{2} + 8 a + 6 + \left(4 a^{2} + 3 a + 4\right)\cdot 11 + \left(8 a^{2} + 10 a + 9\right)\cdot 11^{2} + \left(7 a^{2} + 4\right)\cdot 11^{3} + \left(8 a^{2} + a + 4\right)\cdot 11^{4} + \left(10 a^{2} + 3 a + 10\right)\cdot 11^{5} + \left(a^{2} + 7\right)\cdot 11^{6} + \left(8 a + 1\right)\cdot 11^{7} + \left(10 a^{2} + 6 a + 4\right)\cdot 11^{8} + \left(10 a^{2} + 10 a + 4\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a^{2} + 6 a + 9 + \left(5 a^{2} + 2 a + 1\right)\cdot 11 + \left(3 a^{2} + 2 a + 5\right)\cdot 11^{2} + \left(2 a^{2} + 2 a + 3\right)\cdot 11^{3} + 6 a\cdot 11^{4} + \left(3 a^{2} + 4 a + 8\right)\cdot 11^{5} + \left(5 a^{2} + 6 a\right)\cdot 11^{6} + \left(7 a^{2} + 10 a + 7\right)\cdot 11^{7} + \left(2 a^{2} + 4 a + 5\right)\cdot 11^{8} + \left(5 a^{2} + a + 8\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( a^{2} + 10 + \left(8 a^{2} + 8\right)\cdot 11 + \left(6 a^{2} + 4 a + 10\right)\cdot 11^{2} + \left(4 a^{2} + 7 a + 7\right)\cdot 11^{3} + \left(10 a^{2} + 4 a + 6\right)\cdot 11^{4} + \left(2 a^{2} + 2 a + 3\right)\cdot 11^{5} + \left(6 a^{2} + 2 a + 6\right)\cdot 11^{6} + \left(4 a^{2} + 3 a + 7\right)\cdot 11^{7} + \left(5 a^{2} + 5 a + 1\right)\cdot 11^{8} + \left(2 a^{2} + 10 a + 4\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( a^{2} + 3 a + 10 + \left(9 a^{2} + 7 a + 2\right)\cdot 11 + \left(6 a^{2} + 7 a + 7\right)\cdot 11^{2} + \left(9 a^{2} + 2 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 5 a\right)\cdot 11^{4} + \left(8 a^{2} + 5 a + 7\right)\cdot 11^{5} + \left(2 a^{2} + 8 a + 1\right)\cdot 11^{6} + \left(6 a^{2} + 10 a + 6\right)\cdot 11^{7} + \left(6 a^{2} + 9 a + 10\right)\cdot 11^{8} + \left(8 a^{2} + 4\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a^{2} + 9 a + 10 + \left(9 a^{2} + 6 a + 6\right)\cdot 11 + \left(6 a^{2} + 5 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 4 a + 1\right)\cdot 11^{3} + \left(8 a^{2} + 10 a + 4\right)\cdot 11^{4} + \left(5 a^{2} + 9 a + 4\right)\cdot 11^{5} + \left(5 a^{2} + 6 a + 8\right)\cdot 11^{6} + \left(8 a^{2} + 10 a + 4\right)\cdot 11^{7} + \left(9 a^{2} + 10 a\right)\cdot 11^{8} + \left(3 a^{2} + 4 a + 3\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 9 + 7\cdot 11 + 4\cdot 11^{2} + 5\cdot 11^{3} + 2\cdot 11^{4} + 2\cdot 11^{5} + 5\cdot 11^{6} + 4\cdot 11^{7} + 8\cdot 11^{8} + 3\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 4 a^{2} + 7 a + 1 + \left(6 a^{2} + a + 6\right)\cdot 11 + \left(3 a + 8\right)\cdot 11^{2} + \left(2 a^{2} + 4 a + 6\right)\cdot 11^{3} + \left(2 a^{2} + 5 a + 6\right)\cdot 11^{4} + \left(2 a^{2} + 7 a + 10\right)\cdot 11^{5} + \left(8 a + 4\right)\cdot 11^{6} + \left(6 a^{2} + 1\right)\cdot 11^{7} + \left(9 a^{2} + 6 a\right)\cdot 11^{8} + \left(a^{2} + 4 a + 4\right)\cdot 11^{9} +O(11^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $-2$ |
| $28$ | $2$ | $(1,8)(2,7)(5,6)$ | $0$ |
| $56$ | $3$ | $(1,2,5)(6,8,7)$ | $0$ |
| $42$ | $4$ | $(1,3,4,2)(5,7,6,8)$ | $2$ |
| $56$ | $6$ | $(1,6,2,8,5,7)$ | $0$ |
| $48$ | $7$ | $(1,2,3,8,5,7,4)$ | $-1$ |
| $42$ | $8$ | $(1,8,3,5,4,7,2,6)$ | $0$ |
| $42$ | $8$ | $(1,5,2,8,4,6,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.