Basic invariants
Dimension: | $3$ |
Group: | $\GL(3,2)$ |
Conductor: | \(12241352\)\(\medspace = 2^{3} \cdot 1237^{2} \) |
Artin stem field: | Galois closure of 7.3.97930816.2 |
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 7.3.97930816.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} + 2x^{5} - 12x^{4} + 10x^{3} - 14x^{2} + 10x - 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ | \( 5 a^{2} + 4 a + 1 + \left(9 a^{2} + 15 a\right)\cdot 17 + \left(2 a^{2} + 6 a + 16\right)\cdot 17^{2} + \left(6 a^{2} + 14 a + 4\right)\cdot 17^{3} + \left(11 a^{2} + 12 a + 11\right)\cdot 17^{4} + \left(7 a^{2} + a + 13\right)\cdot 17^{5} + \left(10 a^{2} + 15 a + 5\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 2 }$ | $=$ | \( 12 a^{2} + 10 a + \left(10 a^{2} + a + 1\right)\cdot 17 + \left(2 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(8 a^{2} + 7 a + 11\right)\cdot 17^{3} + \left(3 a^{2} + 11\right)\cdot 17^{4} + \left(16 a^{2} + 16 a + 13\right)\cdot 17^{5} + \left(5 a^{2} + 3 a + 2\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 3 }$ | $=$ | \( 16 + 15\cdot 17 + 12\cdot 17^{2} + 5\cdot 17^{3} + 16\cdot 17^{4} + 9\cdot 17^{5} + 14\cdot 17^{6} +O(17^{7})\) |
$r_{ 4 }$ | $=$ | \( a + 14 + \left(a^{2} + 14 a + 12\right)\cdot 17 + \left(7 a^{2} + 5 a + 8\right)\cdot 17^{2} + \left(4 a^{2} + a + 11\right)\cdot 17^{3} + \left(2 a^{2} + 4 a + 3\right)\cdot 17^{4} + \left(7 a^{2} + 16 a + 4\right)\cdot 17^{5} + \left(10 a + 2\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 5 }$ | $=$ | \( 13 a^{2} + 7 a + \left(4 a^{2} + 15 a + 4\right)\cdot 17 + \left(15 a^{2} + 14 a + 14\right)\cdot 17^{2} + \left(a^{2} + 4 a + 9\right)\cdot 17^{3} + \left(11 a^{2} + 8 a + 9\right)\cdot 17^{4} + \left(4 a^{2} + 7 a + 2\right)\cdot 17^{5} + \left(7 a^{2} + 13 a + 1\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 6 }$ | $=$ | \( 4 a^{2} + 9 a + 11 + \left(11 a^{2} + 4 a + 2\right)\cdot 17 + \left(11 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(10 a^{2} + 10 a + 4\right)\cdot 17^{3} + \left(3 a^{2} + 4 a + 10\right)\cdot 17^{4} + \left(5 a^{2} + 10 a + 8\right)\cdot 17^{5} + \left(9 a^{2} + 9 a + 2\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 7 }$ | $=$ | \( 3 a + 9 + \left(14 a^{2} + 14\right)\cdot 17 + \left(11 a^{2} + 16 a + 10\right)\cdot 17^{2} + \left(2 a^{2} + 11 a + 2\right)\cdot 17^{3} + \left(2 a^{2} + 3 a + 5\right)\cdot 17^{4} + \left(10 a^{2} + 16 a + 15\right)\cdot 17^{5} + \left(14 a + 4\right)\cdot 17^{6} +O(17^{7})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$21$ | $2$ | $(1,6)(5,7)$ | $-1$ |
$56$ | $3$ | $(1,7,4)(2,3,6)$ | $0$ |
$42$ | $4$ | $(2,5)(3,4,6,7)$ | $1$ |
$24$ | $7$ | $(1,7,2,5,3,4,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$24$ | $7$ | $(1,5,6,2,4,7,3)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.