Basic invariants
| Dimension: | $6$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(37884025\)\(\medspace = 5^{2} \cdot 1231^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.37884025.1 |
| 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 7.3.37884025.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{5} - 6x^{4} + x^{3} + 6x^{2} + 7x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{3} + x + 28 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a^{2} + 5 + \left(30 a^{2} + 17 a + 4\right)\cdot 31 + \left(13 a^{2} + 26 a + 20\right)\cdot 31^{2} + \left(24 a^{2} + 25 a + 16\right)\cdot 31^{3} + \left(28 a^{2} + 11 a + 16\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 4\cdot 31 + 23\cdot 31^{2} + 27\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 a^{2} + 29 a + 25 + \left(9 a^{2} + 2 a + 10\right)\cdot 31 + \left(14 a^{2} + 18 a + 20\right)\cdot 31^{2} + \left(20 a^{2} + 10 a + 3\right)\cdot 31^{3} + \left(27 a^{2} + 3 a + 26\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a^{2} + 18 a + \left(26 a^{2} + 25 a + 22\right)\cdot 31 + \left(17 a^{2} + 24 a + 3\right)\cdot 31^{2} + \left(18 a^{2} + 17 a + 13\right)\cdot 31^{3} + \left(22 a + 10\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a^{2} + 20 a + 20 + \left(28 a^{2} + 5 a + 2\right)\cdot 31 + \left(8 a^{2} + 8\right)\cdot 31^{2} + \left(2 a^{2} + 19 a + 2\right)\cdot 31^{3} + \left(22 a^{2} + 3 a + 4\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a^{2} + 24 a + 28 + \left(7 a^{2} + 30 a + 19\right)\cdot 31 + \left(4 a^{2} + 5 a + 25\right)\cdot 31^{2} + \left(10 a^{2} + 25 a + 17\right)\cdot 31^{3} + \left(8 a^{2} + 4 a + 15\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a^{2} + 2 a + 20 + \left(21 a^{2} + 11 a + 28\right)\cdot 31 + \left(2 a^{2} + 17 a + 22\right)\cdot 31^{2} + \left(17 a^{2} + 25 a + 11\right)\cdot 31^{3} + \left(5 a^{2} + 15 a + 11\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,4)(2,3)$ | $2$ |
| $56$ | $3$ | $(2,7,5)(3,4,6)$ | $0$ |
| $42$ | $4$ | $(1,7)(2,4,5,6)$ | $0$ |
| $24$ | $7$ | $(1,7,4,5,6,3,2)$ | $-1$ |
| $24$ | $7$ | $(1,5,2,4,3,7,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.