Basic invariants
Dimension: | $7$ |
Group: | $\GL(3,2)$ |
Conductor: | \(113640397050625\)\(\medspace = 5^{4} \cdot 653^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.3.266505625.2 |
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 7.3.266505625.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 3x^{6} + 3x^{5} + 6x^{4} - 17x^{3} + 6x^{2} + 16x - 8 \) . |
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:
$r_{ 1 }$ | $=$ | \( 3 + 2\cdot 17 + 14\cdot 17^{2} + 3\cdot 17^{3} + 10\cdot 17^{4} + 4\cdot 17^{5} + 15\cdot 17^{6} +O(17^{7})\) |
$r_{ 2 }$ | $=$ | \( 5 a^{2} + 15 a + 9 + \left(8 a^{2} + 3 a + 4\right)\cdot 17 + \left(4 a^{2} + 14 a + 3\right)\cdot 17^{2} + \left(16 a^{2} + 9\right)\cdot 17^{3} + \left(7 a^{2} + 15 a + 13\right)\cdot 17^{4} + \left(7 a^{2} + 4 a + 12\right)\cdot 17^{5} + \left(15 a + 4\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{2} + 2 a + 7 + \left(7 a^{2} + 11 a + 9\right)\cdot 17 + \left(10 a^{2} + 12 a + 1\right)\cdot 17^{2} + \left(3 a^{2} + 8 a + 12\right)\cdot 17^{3} + \left(16 a^{2} + 16 a + 7\right)\cdot 17^{4} + \left(16 a^{2} + 10 a + 13\right)\cdot 17^{5} + \left(15 a^{2} + 3\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 4 }$ | $=$ | \( 10 a^{2} + 1 + \left(a^{2} + 2 a\right)\cdot 17 + \left(2 a^{2} + 7 a + 13\right)\cdot 17^{2} + \left(14 a^{2} + 7 a + 7\right)\cdot 17^{3} + \left(9 a^{2} + 2 a + 3\right)\cdot 17^{4} + \left(9 a^{2} + a + 14\right)\cdot 17^{5} + \left(a + 4\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 5 }$ | $=$ | \( 12 a + \left(13 a^{2} + 7 a + 9\right)\cdot 17 + \left(6 a^{2} + 8 a + 16\right)\cdot 17^{2} + \left(15 a^{2} + 7 a + 4\right)\cdot 17^{3} + \left(2 a^{2} + 3 a + 7\right)\cdot 17^{4} + \left(14 a^{2} + 13 a + 11\right)\cdot 17^{5} + \left(15 a^{2} + 11 a + 6\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 6 }$ | $=$ | \( 3 a^{2} + 16 a + 2 + \left(13 a^{2} + 2 a + 9\right)\cdot 17 + \left(15 a^{2} + 13 a + 5\right)\cdot 17^{2} + \left(a^{2} + 16 a + 7\right)\cdot 17^{3} + \left(8 a^{2} + 16 a + 16\right)\cdot 17^{4} + \left(13 a^{2} + 7 a + 10\right)\cdot 17^{5} + \left(14 a^{2} + 12 a + 11\right)\cdot 17^{6} +O(17^{7})\) |
$r_{ 7 }$ | $=$ | \( 14 a^{2} + 6 a + 15 + \left(7 a^{2} + 6 a + 16\right)\cdot 17 + \left(11 a^{2} + 12 a + 13\right)\cdot 17^{2} + \left(16 a^{2} + 9 a + 5\right)\cdot 17^{3} + \left(5 a^{2} + 13 a + 9\right)\cdot 17^{4} + \left(6 a^{2} + 12 a\right)\cdot 17^{5} + \left(3 a^{2} + 9 a + 4\right)\cdot 17^{6} +O(17^{7})\) |
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 |
$1$ | $1$ | $()$ | $7$ |
$21$ | $2$ | $(1,5)(6,7)$ | $-1$ |
$56$ | $3$ | $(1,3,5)(4,7,6)$ | $1$ |
$42$ | $4$ | $(1,2)(3,4,5,7)$ | $-1$ |
$24$ | $7$ | $(1,7,6,3,4,5,2)$ | $0$ |
$24$ | $7$ | $(1,3,2,6,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.