Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(24026312\)\(\medspace = 2^{3} \cdot 1733^{2} \) |
| Artin stem field: | Galois closure of 7.3.192210496.1 |
| 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 7.3.192210496.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 4x^{5} - 2x^{4} - 4x^{3} - 8x^{2} - 4x - 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{3} + x + 40 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 a + 40 + \left(13 a^{2} + 3 a + 4\right)\cdot 43 + \left(30 a^{2} + 22 a + 21\right)\cdot 43^{2} + \left(21 a^{2} + 21 a\right)\cdot 43^{3} + \left(30 a^{2} + 18\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 a^{2} + 5 a + 15 + \left(16 a^{2} + 34 a + 3\right)\cdot 43 + \left(26 a^{2} + 23 a + 2\right)\cdot 43^{2} + \left(14 a^{2} + 11 a + 32\right)\cdot 43^{3} + \left(13 a^{2} + 15 a + 5\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 40 + 34\cdot 43 + 18\cdot 43^{3} + 16\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a^{2} + 21 a + 31 + \left(24 a^{2} + 20 a + 26\right)\cdot 43 + \left(31 a + 15\right)\cdot 43^{2} + \left(33 a^{2} + 13 a + 22\right)\cdot 43^{3} + \left(24 a^{2} + 4 a + 28\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a^{2} + 15 a + 1 + \left(12 a^{2} + 14 a + 29\right)\cdot 43 + \left(26 a^{2} + 3 a + 30\right)\cdot 43^{2} + \left(27 a^{2} + 14 a + 40\right)\cdot 43^{3} + \left(31 a^{2} + 40 a + 17\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a^{2} + 35 a + 6 + \left(5 a^{2} + 18 a\right)\cdot 43 + \left(12 a^{2} + 32 a + 9\right)\cdot 43^{2} + \left(31 a^{2} + 7 a + 21\right)\cdot 43^{3} + \left(30 a^{2} + 38 a + 32\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 31 a^{2} + 23 a + 39 + \left(13 a^{2} + 37 a + 29\right)\cdot 43 + \left(33 a^{2} + 15 a + 6\right)\cdot 43^{2} + \left(17 a + 37\right)\cdot 43^{3} + \left(41 a^{2} + 30 a + 9\right)\cdot 43^{4} +O(43^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,7)(2,4)$ | $-1$ |
| $56$ | $3$ | $(1,2,5)(3,7,6)$ | $0$ |
| $42$ | $4$ | $(2,3,5,7)(4,6)$ | $1$ |
| $24$ | $7$ | $(1,7,4,6,2,3,5)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ | $7$ | $(1,6,5,4,3,7,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.