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 7.7.18424261696.2 |
| 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 7.7.18424261696.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 9x^{5} + 35x^{4} - 11x^{3} - 37x^{2} + 9x + 11 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 24 a^{2} + 16 a + 3 + \left(27 a^{2} + 8 a + 25\right)\cdot 37 + \left(27 a^{2} + 23 a + 28\right)\cdot 37^{2} + \left(6 a^{2} + 36 a + 5\right)\cdot 37^{3} + \left(13 a + 18\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 7\cdot 37 + 6\cdot 37^{2} + 26\cdot 37^{3} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a^{2} + 29 a + 36 + \left(29 a^{2} + 2 a + 29\right)\cdot 37 + \left(15 a^{2} + 3 a + 7\right)\cdot 37^{2} + \left(34 a^{2} + 3 a + 2\right)\cdot 37^{3} + \left(12 a^{2} + 7 a + 34\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 a^{2} + 7 a + 6 + \left(18 a^{2} + 14 a + 27\right)\cdot 37 + \left(32 a^{2} + 19 a + 10\right)\cdot 37^{2} + \left(14 a^{2} + 14 a + 1\right)\cdot 37^{3} + \left(a^{2} + 31 a + 23\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a^{2} + 14 a + 6 + \left(3 a^{2} + 34\right)\cdot 37 + \left(6 a^{2} + 12 a + 5\right)\cdot 37^{2} + \left(2 a + 13\right)\cdot 37^{3} + \left(3 a^{2} + 28 a + 31\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a^{2} + 14 a + 8 + \left(27 a^{2} + 14 a + 24\right)\cdot 37 + \left(13 a^{2} + 31 a + 9\right)\cdot 37^{2} + \left(15 a^{2} + 22 a + 3\right)\cdot 37^{3} + \left(35 a^{2} + 28 a + 11\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 a^{2} + 31 a + 3 + \left(3 a^{2} + 33 a\right)\cdot 37 + \left(15 a^{2} + 21 a + 5\right)\cdot 37^{2} + \left(2 a^{2} + 31 a + 22\right)\cdot 37^{3} + \left(21 a^{2} + a + 29\right)\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $21$ | $2$ | $(2,6)(3,7)$ | $0$ |
| $56$ | $3$ | $(1,7,3)(2,5,6)$ | $-1$ |
| $42$ | $4$ | $(1,4)(2,7,6,3)$ | $0$ |
| $24$ | $7$ | $(1,2,5,7,6,3,4)$ | $1$ |
| $24$ | $7$ | $(1,7,4,5,3,2,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.