Basic invariants
Dimension: | $3$ |
Group: | $\GL(3,2)$ |
Conductor: | \(3495368\)\(\medspace = 2^{3} \cdot 661^{2} \) |
Artin stem field: | Galois closure of 7.3.27962944.2 |
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 7.3.27962944.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} + 3x^{4} - 5x^{3} + 7x^{2} - 5x + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 11 a^{2} + 24 a + 18 + \left(19 a^{2} + 21 a + 2\right)\cdot 29 + \left(22 a^{2} + 24 a + 4\right)\cdot 29^{2} + \left(11 a^{2} + 28 a + 26\right)\cdot 29^{3} + \left(24 a^{2} + 15 a + 11\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 18 + 3\cdot 29 + 10\cdot 29^{2} + 2\cdot 29^{3} + 12\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 25 a^{2} + 16 a + 5 + \left(13 a^{2} + 21 a + 21\right)\cdot 29 + \left(14 a^{2} + 19 a + 22\right)\cdot 29^{2} + \left(3 a^{2} + a + 12\right)\cdot 29^{3} + \left(a^{2} + 9 a + 8\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 19 a^{2} + 12 a + 26 + \left(17 a^{2} + 8 a + 6\right)\cdot 29 + \left(2 a^{2} + 25 a + 26\right)\cdot 29^{2} + \left(16 a^{2} + 16 a + 19\right)\cdot 29^{3} + \left(13 a^{2} + 19 a + 5\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 21 a^{2} + 9 a + 12 + \left(5 a^{2} + 19 a + 13\right)\cdot 29 + \left(8 a^{2} + 5 a + 23\right)\cdot 29^{2} + \left(22 a^{2} + 20\right)\cdot 29^{3} + \left(7 a^{2} + 11 a + 18\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 6 }$ | $=$ | \( 26 a^{2} + 25 a + 9 + \left(3 a^{2} + 16 a + 1\right)\cdot 29 + \left(27 a^{2} + 27 a + 10\right)\cdot 29^{2} + \left(23 a^{2} + 28 a + 13\right)\cdot 29^{3} + \left(25 a^{2} + a + 23\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 7 }$ | $=$ | \( 14 a^{2} + a + \left(26 a^{2} + 28 a + 9\right)\cdot 29 + \left(11 a^{2} + 12 a + 19\right)\cdot 29^{2} + \left(9 a^{2} + 10 a + 20\right)\cdot 29^{3} + \left(14 a^{2} + 6\right)\cdot 29^{4} +O(29^{5})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$21$ | $2$ | $(1,7)(2,4)$ | $-1$ |
$56$ | $3$ | $(1,5,4)(2,3,6)$ | $0$ |
$42$ | $4$ | $(1,6,5,2)(3,7)$ | $1$ |
$24$ | $7$ | $(1,6,5,4,2,7,3)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$24$ | $7$ | $(1,4,3,5,7,6,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.