Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(1279091\)\(\medspace = 11^{3} \cdot 31^{2} \) |
| Artin stem field: | Galois closure of 7.3.1702470121.1 |
| 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 7.3.1702470121.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 11x^{5} - 22x^{4} + 33x^{2} + 33x + 11 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 3 a + 23 + \left(15 a^{2} + 9 a + 15\right)\cdot 29 + \left(6 a^{2} + 8 a + 7\right)\cdot 29^{2} + \left(25 a^{2} + 27 a + 8\right)\cdot 29^{3} + \left(5 a^{2} + 22 a + 20\right)\cdot 29^{4} + \left(a^{2} + 13 a + 10\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a^{2} + 28 a + 8 + \left(25 a^{2} + 8 a + 15\right)\cdot 29 + \left(4 a^{2} + 23 a + 21\right)\cdot 29^{2} + \left(18 a^{2} + 28 a + 15\right)\cdot 29^{3} + \left(26 a^{2} + 19 a\right)\cdot 29^{4} + \left(15 a^{2} + 2 a + 18\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a^{2} + 28 a + 19 + \left(21 a^{2} + 22 a + 19\right)\cdot 29 + \left(2 a^{2} + 2 a + 18\right)\cdot 29^{2} + \left(a^{2} + 25 a + 2\right)\cdot 29^{3} + \left(5 a^{2} + 18 a + 20\right)\cdot 29^{4} + \left(2 a^{2} + 9 a + 28\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 12 + \left(17 a^{2} + 25 a + 18\right)\cdot 29 + \left(21 a^{2} + 4 a + 27\right)\cdot 29^{2} + \left(17 a^{2} + 5 a + 7\right)\cdot 29^{3} + \left(12 a^{2} + 17 a\right)\cdot 29^{4} + \left(13 a^{2} + 9 a + 27\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 10\cdot 29 + 16\cdot 29^{2} + 14\cdot 29^{3} + 9\cdot 29^{4} + 11\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 2 a + 19 + \left(11 a^{2} + 26 a + 25\right)\cdot 29 + \left(21 a^{2} + 2 a + 4\right)\cdot 29^{2} + \left(9 a^{2} + 4 a + 14\right)\cdot 29^{3} + \left(26 a^{2} + 19 a + 19\right)\cdot 29^{4} + \left(10 a^{2} + 16 a + 1\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a^{2} + 1 + \left(25 a^{2} + 23 a + 11\right)\cdot 29 + \left(15 a + 19\right)\cdot 29^{2} + \left(15 a^{2} + 25 a + 23\right)\cdot 29^{3} + \left(10 a^{2} + 17 a + 16\right)\cdot 29^{4} + \left(14 a^{2} + 5 a + 18\right)\cdot 29^{5} +O(29^{6})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $56$ | $3$ | $(1,2,7)(3,5,4)$ | $0$ |
| $42$ | $4$ | $(1,3,7,5)(2,6)$ | $1$ |
| $24$ | $7$ | $(1,4,3,2,6,7,5)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ | $7$ | $(1,2,5,3,7,4,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.