Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(19669225\)\(\medspace = 5^{2} \cdot 887^{2} \) |
| Artin stem field: | Galois closure of 7.3.491730625.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $\PSL(2,7)$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $\GL(3,2)$ |
| Projective stem field: | Galois closure of 7.3.491730625.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - x^{5} - 15x^{4} - 11x^{2} + x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 5 a + 7 + \left(7 a^{2} + a + 7\right)\cdot 11 + \left(4 a^{2} + a + 5\right)\cdot 11^{2} + \left(5 a^{2} + 10 a + 7\right)\cdot 11^{3} + \left(10 a^{2} + 9 a + 2\right)\cdot 11^{4} + \left(8 a^{2} + 6 a + 8\right)\cdot 11^{5} + \left(8 a^{2} + 5 a + 10\right)\cdot 11^{6} + \left(10 a^{2} + 3 a + 3\right)\cdot 11^{7} +O(11^{8})\)
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| $r_{ 2 }$ | $=$ |
\( a^{2} + 5 a + 9 + \left(5 a^{2} + 5 a + 4\right)\cdot 11 + \left(9 a^{2} + 8 a + 1\right)\cdot 11^{2} + \left(3 a + 10\right)\cdot 11^{3} + \left(8 a^{2} + 4 a\right)\cdot 11^{4} + \left(10 a^{2} + 2 a\right)\cdot 11^{5} + \left(9 a^{2} + 6 a + 8\right)\cdot 11^{6} + \left(10 a^{2} + 10 a + 7\right)\cdot 11^{7} +O(11^{8})\)
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| $r_{ 3 }$ | $=$ |
\( 3 a^{2} + 2 a + 8 + \left(9 a^{2} + a + 6\right)\cdot 11 + \left(a^{2} + 7 a + 9\right)\cdot 11^{2} + \left(5 a^{2} + 4 a + 4\right)\cdot 11^{3} + \left(9 a^{2} + 5 a + 6\right)\cdot 11^{4} + \left(4 a^{2} + 8 a + 10\right)\cdot 11^{5} + \left(10 a^{2} + 2 a + 4\right)\cdot 11^{6} + \left(a^{2} + 6 a + 10\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a^{2} + 2 a + 6 + \left(a^{2} + 2 a + 3\right)\cdot 11 + \left(a^{2} + 8\right)\cdot 11^{2} + \left(2 a^{2} + 10 a + 6\right)\cdot 11^{3} + \left(8 a^{2} + a + 10\right)\cdot 11^{4} + \left(6 a^{2} + 2 a + 8\right)\cdot 11^{5} + \left(3 a^{2} + 8 a + 3\right)\cdot 11^{6} + \left(8 a^{2} + 4 a + 4\right)\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 2\cdot 11^{2} + 5\cdot 11^{3} + 8\cdot 11^{4} + 9\cdot 11^{5} + 7\cdot 11^{6} + 7\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a^{2} + 4 a + 4 + \left(2 a^{2} + 7 a + 5\right)\cdot 11 + \left(5 a^{2} + 9 a + 6\right)\cdot 11^{2} + \left(3 a^{2} + a + 8\right)\cdot 11^{3} + \left(3 a^{2} + 10 a + 7\right)\cdot 11^{4} + \left(6 a^{2} + a + 4\right)\cdot 11^{5} + \left(9 a^{2} + 8 a + 4\right)\cdot 11^{6} + \left(2 a^{2} + 2 a + 4\right)\cdot 11^{7} +O(11^{8})\)
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| $r_{ 7 }$ | $=$ |
\( 7 a^{2} + 4 a + 6 + \left(7 a^{2} + 4 a + 4\right)\cdot 11 + \left(10 a^{2} + 6 a + 10\right)\cdot 11^{2} + \left(4 a^{2} + 2 a\right)\cdot 11^{3} + \left(4 a^{2} + a + 7\right)\cdot 11^{4} + \left(6 a^{2} + 1\right)\cdot 11^{5} + \left(a^{2} + 2 a + 4\right)\cdot 11^{6} + \left(9 a^{2} + 5 a + 5\right)\cdot 11^{7} +O(11^{8})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $56$ | $3$ | $(1,4,6)(2,3,7)$ | $0$ |
| $42$ | $4$ | $(1,2,4,3)(5,6)$ | $1$ |
| $24$ | $7$ | $(1,2,4,5,6,3,7)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ | $7$ | $(1,5,7,4,3,2,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.