Basic invariants
Dimension: | $3$ |
Group: | $\GL(3,2)$ |
Conductor: | \(1670792\)\(\medspace = 2^{3} \cdot 457^{2} \) |
Artin stem field: | Galois closure of 7.3.13366336.2 |
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 7.3.13366336.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} - 2x^{5} + 6x^{4} - 4x^{3} - 2x^{2} + 4x - 2 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 22 a^{2} + 2 a + 2 + \left(23 a^{2} + 25 a + 21\right)\cdot 31 + \left(24 a + 1\right)\cdot 31^{2} + \left(4 a^{2} + 21 a + 1\right)\cdot 31^{3} + \left(19 a^{2} + 27 a + 14\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 8 a^{2} + 17 a + 3 + \left(22 a^{2} + 8 a + 20\right)\cdot 31 + \left(30 a^{2} + 18 a + 21\right)\cdot 31^{2} + \left(24 a^{2} + 4 a + 4\right)\cdot 31^{3} + \left(5 a^{2} + 14 a + 5\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( a^{2} + 12 a + 19 + \left(16 a^{2} + 28 a + 5\right)\cdot 31 + \left(30 a^{2} + 18 a + 11\right)\cdot 31^{2} + \left(a^{2} + 4 a + 20\right)\cdot 31^{3} + \left(6 a^{2} + 20 a + 15\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 24 + 8\cdot 31 + 10\cdot 31^{2} + 4\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 2 a^{2} + 4 a + 17 + \left(18 a^{2} + 16 a + 24\right)\cdot 31 + \left(15 a^{2} + 11 a + 5\right)\cdot 31^{2} + \left(30 a^{2} + 10\right)\cdot 31^{3} + \left(5 a^{2} + 6 a + 10\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 15 a^{2} + 23 a + 5 + \left(29 a^{2} + 20 a + 1\right)\cdot 31 + \left(a^{2} + a + 7\right)\cdot 31^{2} + \left(27 a^{2} + 3 a + 18\right)\cdot 31^{3} + \left(25 a^{2} + 13 a + 23\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 14 a^{2} + 4 a + 25 + \left(14 a^{2} + 25 a + 11\right)\cdot 31 + \left(13 a^{2} + 17 a + 4\right)\cdot 31^{2} + \left(4 a^{2} + 27 a + 3\right)\cdot 31^{3} + \left(30 a^{2} + 11 a + 16\right)\cdot 31^{4} +O(31^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$21$ | $2$ | $(4,7)(5,6)$ | $-1$ |
$56$ | $3$ | $(2,6,5)(3,7,4)$ | $0$ |
$42$ | $4$ | $(1,2)(4,6,7,5)$ | $1$ |
$24$ | $7$ | $(1,2,4,3,6,7,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$24$ | $7$ | $(1,3,5,4,7,2,6)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.