Basic invariants
Dimension: | $6$ |
Group: | $\GL(3,2)$ |
Conductor: | \(1814078464\)\(\medspace = 2^{10} \cdot 11^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.3.1814078464.1 |
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 7.3.1814078464.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 2x^{5} - 12x^{4} - 14x^{3} + 10x^{2} + 10x - 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{3} + 4x + 17 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 6 a^{2} + 18 a + 1 + \left(13 a^{2} + 12 a + 13\right)\cdot 19 + \left(17 a^{2} + 17 a + 17\right)\cdot 19^{2} + \left(3 a^{2} + 15 a + 8\right)\cdot 19^{3} + \left(4 a^{2} + 17 a + 16\right)\cdot 19^{4} + \left(9 a^{2} + 16 a + 7\right)\cdot 19^{5} + \left(5 a^{2} + 17 a + 9\right)\cdot 19^{6} +O(19^{7})\)
$r_{ 2 }$ |
$=$ |
\( 12 a^{2} + 9 a + 17 + \left(9 a^{2} + 5 a\right)\cdot 19 + \left(17 a^{2} + 9 a + 10\right)\cdot 19^{2} + \left(2 a^{2} + 3 a + 9\right)\cdot 19^{3} + \left(3 a^{2} + 2 a + 6\right)\cdot 19^{4} + \left(17 a^{2} + 16 a + 5\right)\cdot 19^{5} + \left(11 a^{2} + 14 a + 5\right)\cdot 19^{6} +O(19^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 13 a^{2} + 8 a + 7 + \left(10 a^{2} + 12 a + 16\right)\cdot 19 + \left(a^{2} + 8 a + 11\right)\cdot 19^{2} + \left(17 a^{2} + 12 a + 15\right)\cdot 19^{3} + \left(9 a^{2} + 11 a + 11\right)\cdot 19^{4} + \left(9 a^{2} + a + 16\right)\cdot 19^{5} + \left(13 a^{2} + 17 a + 15\right)\cdot 19^{6} +O(19^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 15 + 8\cdot 19 + 8\cdot 19^{2} + 18\cdot 19^{3} + 8\cdot 19^{4} +O(19^{7})\)
| $r_{ 5 }$ |
$=$ |
\( 9 a^{2} + 8 a + 9 + \left(4 a^{2} + 18 a + 8\right)\cdot 19 + \left(6 a^{2} + 18 a + 12\right)\cdot 19^{2} + \left(3 a^{2} + 12 a + 13\right)\cdot 19^{3} + \left(13 a^{2} + 8\right)\cdot 19^{4} + \left(4 a^{2} + 5 a + 8\right)\cdot 19^{5} + \left(a^{2} + 15 a + 4\right)\cdot 19^{6} +O(19^{7})\)
| $r_{ 6 }$ |
$=$ |
\( 4 a^{2} + 12 a + 2 + \left(a^{2} + 6 a + 6\right)\cdot 19 + \left(14 a^{2} + a + 14\right)\cdot 19^{2} + \left(11 a^{2} + 9 a + 10\right)\cdot 19^{3} + \left(a^{2} + 9\right)\cdot 19^{4} + \left(5 a^{2} + 16 a + 9\right)\cdot 19^{5} + \left(12 a^{2} + 4 a + 8\right)\cdot 19^{6} +O(19^{7})\)
| $r_{ 7 }$ |
$=$ |
\( 13 a^{2} + 2 a + 7 + \left(17 a^{2} + a + 3\right)\cdot 19 + \left(18 a^{2} + a + 1\right)\cdot 19^{2} + \left(17 a^{2} + 3 a + 18\right)\cdot 19^{3} + \left(5 a^{2} + 5 a + 13\right)\cdot 19^{4} + \left(11 a^{2} + a + 8\right)\cdot 19^{5} + \left(12 a^{2} + 6 a + 13\right)\cdot 19^{6} +O(19^{7})\)
| |
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 |
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(2,6)(3,4)$ | $2$ |
$56$ | $3$ | $(1,4,5)(2,7,3)$ | $0$ |
$42$ | $4$ | $(1,7,4,2)(5,6)$ | $0$ |
$24$ | $7$ | $(1,7,3,4,6,5,2)$ | $-1$ |
$24$ | $7$ | $(1,4,2,3,5,7,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.