Basic invariants
Dimension: | $7$ |
Group: | $\GL(3,2)$ |
Conductor: | \(78018073190656\)\(\medspace = 2^{8} \cdot 743^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.3.35331136.2 |
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 7.3.35331136.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} + 5x^{4} - 6x^{3} + 4x^{2} + 8x - 8 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 6 a^{2} + 5 a + 13 + \left(19 a^{2} + 17 a + 3\right)\cdot 29 + \left(4 a^{2} + 23\right)\cdot 29^{2} + \left(20 a^{2} + 25 a + 11\right)\cdot 29^{3} + \left(5 a^{2} + 5 a + 6\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$ |
$=$ |
\( 8 a^{2} + 13 a + 6 + \left(16 a^{2} + 28 a + 9\right)\cdot 29 + \left(25 a^{2} + 17 a + 12\right)\cdot 29^{2} + \left(11 a^{2} + 18 a + 10\right)\cdot 29^{3} + \left(4 a + 28\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 12 a^{2} + 9 a + 25 + \left(15 a^{2} + 13 a + 27\right)\cdot 29 + \left(5 a^{2} + 18 a + 14\right)\cdot 29^{2} + \left(6 a^{2} + 26 a + 10\right)\cdot 29^{3} + \left(4 a^{2} + 11 a + 25\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 15 a^{2} + 11 a + 25 + \left(22 a^{2} + 12 a + 7\right)\cdot 29 + \left(27 a^{2} + 10 a + 15\right)\cdot 29^{2} + \left(25 a^{2} + 14 a + 19\right)\cdot 29^{3} + \left(22 a^{2} + 18 a + 19\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 17 + 15\cdot 29 + 13\cdot 29^{2} + 9\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 20 a^{2} + 22 a + 26 + \left(17 a^{2} + 21 a + 1\right)\cdot 29 + \left(25 a^{2} + 20 a + 3\right)\cdot 29^{2} + \left(11 a^{2} + a + 18\right)\cdot 29^{3} + \left(7 a^{2} + 23 a\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 26 a^{2} + 27 a + 5 + \left(24 a^{2} + 22 a + 21\right)\cdot 29 + \left(26 a^{2} + 18 a + 4\right)\cdot 29^{2} + \left(10 a^{2} + 7\right)\cdot 29^{3} + \left(17 a^{2} + 23 a + 4\right)\cdot 29^{4} +O(29^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $7$ |
$21$ | $2$ | $(1,7)(2,6)$ | $-1$ |
$56$ | $3$ | $(1,3,5)(4,7,6)$ | $1$ |
$42$ | $4$ | $(1,6,5,4)(2,3)$ | $-1$ |
$24$ | $7$ | $(1,7,6,3,2,5,4)$ | $0$ |
$24$ | $7$ | $(1,3,4,6,5,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.