Basic invariants
Dimension: | $6$ |
Group: | $\GL(3,2)$ |
Conductor: | \(28291761\)\(\medspace = 3^{6} \cdot 197^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.3.28291761.2 |
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 7.3.28291761.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 3x^{5} - 6x^{4} + 3x^{3} - 6x^{2} + 5x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{3} + 2x + 11 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 6 a^{2} + 10 a + 5 + \left(12 a^{2} + 4 a + 11\right)\cdot 13 + \left(9 a^{2} + 6 a + 9\right)\cdot 13^{2} + \left(12 a^{2} + 9 a + 12\right)\cdot 13^{3} + \left(11 a + 8\right)\cdot 13^{4} + \left(9 a^{2} + 2\right)\cdot 13^{5} +O(13^{6})\)
$r_{ 2 }$ |
$=$ |
\( 4 a^{2} + 8 a + 1 + \left(10 a^{2} + a + 6\right)\cdot 13 + \left(3 a^{2} + 4 a + 12\right)\cdot 13^{2} + \left(a^{2} + 6 a + 12\right)\cdot 13^{3} + \left(4 a^{2} + 3 a + 11\right)\cdot 13^{4} + \left(8 a^{2} + 9 a\right)\cdot 13^{5} +O(13^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 11 + 12\cdot 13 + 5\cdot 13^{3} + 9\cdot 13^{4} + 6\cdot 13^{5} +O(13^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 8 a^{2} + 2 + \left(5 a + 6\right)\cdot 13 + \left(3 a^{2} + 11\right)\cdot 13^{2} + \left(2 a^{2} + 3 a + 9\right)\cdot 13^{3} + \left(9 a^{2} + 4 a + 5\right)\cdot 13^{4} + \left(2 a^{2} + 2 a + 6\right)\cdot 13^{5} +O(13^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a^{2} + 4 a + 8 + \left(4 a^{2} + 5 a\right)\cdot 13 + \left(10 a^{2} + 5 a + 6\right)\cdot 13^{2} + \left(11 a^{2} + 7 a + 11\right)\cdot 13^{3} + \left(7 a^{2} + 10 a\right)\cdot 13^{4} + \left(4 a^{2} + 6 a + 1\right)\cdot 13^{5} +O(13^{6})\)
| $r_{ 6 }$ |
$=$ |
\( a^{2} + 5 a + 10 + \left(2 a^{2} + 6 a + 3\right)\cdot 13 + \left(6 a^{2} + 8 a + 11\right)\cdot 13^{2} + \left(9 a^{2} + 3 a + 10\right)\cdot 13^{3} + \left(12 a^{2} + 5 a + 1\right)\cdot 13^{4} + \left(a^{2} + a + 1\right)\cdot 13^{5} +O(13^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 2 a^{2} + 12 a + 4 + \left(9 a^{2} + 2 a + 11\right)\cdot 13 + \left(5 a^{2} + a + 12\right)\cdot 13^{2} + \left(a^{2} + 9 a + 1\right)\cdot 13^{3} + \left(4 a^{2} + 3 a\right)\cdot 13^{4} + \left(12 a^{2} + 5 a + 7\right)\cdot 13^{5} +O(13^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(2,6)(3,4)$ | $2$ |
$56$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
$42$ | $4$ | $(1,7)(2,4,6,3)$ | $0$ |
$24$ | $7$ | $(1,2,5,4,6,3,7)$ | $-1$ |
$24$ | $7$ | $(1,4,7,5,3,2,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.