Basic invariants
| Dimension: | $8$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(235078545043416064\)\(\medspace = 2^{10} \cdot 19^{6} \cdot 47^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.18424261696.1 |
| Galois orbit size: | $1$ |
| 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 7.7.18424261696.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 12x^{5} + 31x^{3} - 14x^{2} - 8x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{3} + 6x + 35 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{2} + 27 a + 14 + \left(34 a^{2} + 3 a + 28\right)\cdot 37 + \left(35 a^{2} + 2 a + 17\right)\cdot 37^{2} + \left(19 a^{2} + 12 a + 34\right)\cdot 37^{3} + \left(11 a^{2} + 32 a + 3\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a^{2} + 21 + \left(25 a^{2} + 15 a + 16\right)\cdot 37 + \left(17 a^{2} + 11 a + 4\right)\cdot 37^{2} + \left(7 a^{2} + 15 a + 22\right)\cdot 37^{3} + \left(3 a^{2} + 32 a + 23\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a^{2} + 31 a + 35 + \left(34 a^{2} + 29 a + 17\right)\cdot 37 + \left(21 a^{2} + 17 a + 21\right)\cdot 37^{2} + \left(33 a^{2} + 23 a + 15\right)\cdot 37^{3} + \left(20 a^{2} + 28 a + 20\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 22\cdot 37 + 21\cdot 37^{2} + 11\cdot 37^{3} + 20\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a^{2} + 35 a + 22 + \left(6 a^{2} + 26 a + 27\right)\cdot 37 + \left(20 a^{2} + 4 a + 28\right)\cdot 37^{2} + \left(18 a^{2} + 35 a + 28\right)\cdot 37^{3} + \left(21 a^{2} + 27 a + 6\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 a^{2} + 12 a + 7 + \left(33 a^{2} + 6 a + 27\right)\cdot 37 + \left(17 a^{2} + 30 a + 19\right)\cdot 37^{2} + \left(35 a^{2} + 26 a + 22\right)\cdot 37^{3} + \left(3 a^{2} + 13 a + 10\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 30 a^{2} + 6 a + 23 + \left(13 a^{2} + 29 a + 7\right)\cdot 37 + \left(34 a^{2} + 7 a + 34\right)\cdot 37^{2} + \left(32 a^{2} + 35 a + 12\right)\cdot 37^{3} + \left(12 a^{2} + 12 a + 25\right)\cdot 37^{4} +O(37^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $8$ |
| $21$ | $2$ | $(1,4)(5,6)$ | $0$ |
| $56$ | $3$ | $(1,2,4)(3,5,6)$ | $-1$ |
| $42$ | $4$ | $(1,6,4,5)(2,7)$ | $0$ |
| $24$ | $7$ | $(1,7,2,6,3,4,5)$ | $1$ |
| $24$ | $7$ | $(1,6,5,2,4,7,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.