Basic invariants
| Dimension: | $6$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(27290176\)\(\medspace = 2^{6} \cdot 653^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.27290176.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\GL(3,2)$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $\GL(3,2)$ |
| Projective stem field: | Galois closure of 7.3.27290176.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + x^{5} - 3x^{4} + 2x^{3} + 6x^{2} - 6x + 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a^{2} + 26 a + 28 + \left(10 a^{2} + 17 a + 11\right)\cdot 29 + \left(21 a^{2} + 22 a + 2\right)\cdot 29^{2} + \left(15 a^{2} + 23 a + 7\right)\cdot 29^{3} + \left(12 a^{2} + 11 a + 18\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 27\cdot 29^{2} + 4\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a^{2} + 6 a + 18 + \left(20 a + 11\right)\cdot 29 + \left(17 a^{2} + 4 a + 20\right)\cdot 29^{2} + \left(3 a^{2} + 15 a + 8\right)\cdot 29^{3} + \left(3 a^{2} + 9 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( a^{2} + 21 a + 2 + \left(5 a^{2} + 25 a + 5\right)\cdot 29 + \left(4 a^{2} + 4 a + 18\right)\cdot 29^{2} + \left(3 a^{2} + 14 a + 9\right)\cdot 29^{3} + \left(5 a^{2} + 14 a + 8\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a^{2} + 11 a + 1 + \left(13 a^{2} + 14 a + 7\right)\cdot 29 + \left(3 a^{2} + a + 17\right)\cdot 29^{2} + \left(10 a^{2} + 20 a + 28\right)\cdot 29^{3} + \left(11 a^{2} + 2 a + 6\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a^{2} + 25 a + 1 + \left(15 a^{2} + 24 a + 3\right)\cdot 29 + \left(10 a^{2} + 11 a + 2\right)\cdot 29^{2} + \left(7 a^{2} + 22 a + 4\right)\cdot 29^{3} + \left(25 a^{2} + 4 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 27 a^{2} + 27 a + 20 + \left(12 a^{2} + 12 a + 18\right)\cdot 29 + \left(a^{2} + 12 a + 28\right)\cdot 29^{2} + \left(18 a^{2} + 20 a + 27\right)\cdot 29^{3} + \left(14 a + 26\right)\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,3)(2,7)$ | $2$ |
| $56$ | $3$ | $(1,2,6)(3,4,5)$ | $0$ |
| $42$ | $4$ | $(2,5,6,3)(4,7)$ | $0$ |
| $24$ | $7$ | $(1,3,7,4,2,5,6)$ | $-1$ |
| $24$ | $7$ | $(1,4,6,7,5,3,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.