Basic invariants
Dimension: | $3$ |
Group: | $\GL(3,2)$ |
Conductor: | \(1670792\)\(\medspace = 2^{3} \cdot 457^{2} \) |
Artin stem field: | Galois closure of 7.3.13366336.1 |
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 7.3.13366336.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 2x^{5} + 2x^{4} - 4x^{3} + 4x^{2} - 4 \) . |
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:
$r_{ 1 }$ | $=$ |
\( a^{2} + 9 a + \left(25 a^{2} + 19 a + 23\right)\cdot 31 + \left(12 a^{2} + 25 a + 16\right)\cdot 31^{2} + \left(25 a^{2} + 20 a + 19\right)\cdot 31^{3} + \left(24 a^{2} + 6 a + 19\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 11 a^{2} + 2 a + 7 + \left(28 a^{2} + 9 a + 14\right)\cdot 31 + \left(25 a^{2} + 9 a + 17\right)\cdot 31^{2} + \left(21 a^{2} + 6 a + 28\right)\cdot 31^{3} + \left(21 a^{2} + 14 a + 22\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 13 a^{2} + 23 a + 29 + \left(14 a^{2} + 20 a + 4\right)\cdot 31 + \left(a + 21\right)\cdot 31^{2} + \left(21 a^{2} + 26 a + 17\right)\cdot 31^{3} + \left(30 a^{2} + 14 a + 18\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 7 a^{2} + 6 a + 25 + \left(19 a^{2} + a + 28\right)\cdot 31 + \left(4 a^{2} + 20 a + 23\right)\cdot 31^{2} + \left(19 a^{2} + 29 a + 26\right)\cdot 31^{3} + \left(9 a^{2} + a + 14\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 17 a^{2} + 28 a + 21 + \left(a + 6\right)\cdot 31 + \left(29 a^{2} + 7 a + 17\right)\cdot 31^{2} + \left(27 a^{2} + 2 a\right)\cdot 31^{3} + \left(6 a^{2} + 21 a + 18\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 5 + 26\cdot 31 + 5\cdot 31^{2} + 12\cdot 31^{3} + 27\cdot 31^{4} +O(31^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 13 a^{2} + 25 a + 8 + \left(5 a^{2} + 9 a + 20\right)\cdot 31 + \left(20 a^{2} + 29 a + 21\right)\cdot 31^{2} + \left(8 a^{2} + 7 a + 18\right)\cdot 31^{3} + \left(30 a^{2} + 3 a + 2\right)\cdot 31^{4} +O(31^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$21$ | $2$ | $(1,7)(5,6)$ | $-1$ |
$56$ | $3$ | $(1,3,2)(4,7,5)$ | $0$ |
$42$ | $4$ | $(1,4,2,5)(3,6)$ | $1$ |
$24$ | $7$ | $(1,7,4,2,5,3,6)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$24$ | $7$ | $(1,2,6,4,3,7,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.