Basic invariants
Dimension: | $2$ |
Group: | $D_{9}$ |
Conductor: | \(1999\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.15968023992001.1 |
Galois orbit size: | $3$ |
Smallest permutation container: | $D_{9}$ |
Parity: | odd |
Determinant: | 1.1999.2t1.a.a |
Projective image: | $D_9$ |
Projective stem field: | Galois closure of 9.1.15968023992001.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 3x^{8} - 2x^{7} + 3x^{6} + 19x^{5} + 39x^{4} + 85x^{3} + 66x^{2} + 108x + 27 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{3} + x + 35 \)
Roots:
$r_{ 1 }$ | $=$ | \( 3 a^{2} + 14 a + 22 + \left(24 a^{2} + 31 a + 37\right)\cdot 41 + \left(3 a^{2} + 36 a + 28\right)\cdot 41^{2} + \left(23 a + 18\right)\cdot 41^{3} + \left(9 a^{2} + 12 a + 2\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 26 a^{2} + 32 a + 37 + \left(3 a^{2} + 40 a + 15\right)\cdot 41 + \left(11 a^{2} + 14 a + 8\right)\cdot 41^{2} + \left(26 a^{2} + 32 a\right)\cdot 41^{3} + \left(20 a^{2} + 22 a + 15\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 15 a^{2} + 15 a + 16 + \left(24 a^{2} + 34 a + 2\right)\cdot 41 + \left(12 a^{2} + 30 a + 23\right)\cdot 41^{2} + \left(2 a^{2} + 37 a + 11\right)\cdot 41^{3} + \left(15 a^{2} + 13 a + 11\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 35 a + 6 + \left(13 a^{2} + 6 a + 22\right)\cdot 41 + \left(17 a^{2} + 36 a + 12\right)\cdot 41^{2} + \left(12 a^{2} + 11 a + 18\right)\cdot 41^{3} + \left(5 a^{2} + 4 a + 18\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 22 a^{2} + 2 a + 17 + \left(34 a^{2} + 26 a + 15\right)\cdot 41 + \left(2 a^{2} + 4 a + 15\right)\cdot 41^{2} + \left(19 a^{2} + 24 a + 11\right)\cdot 41^{3} + \left(31 a^{2} + 12 a + 23\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 6 }$ | $=$ | \( 32 a^{2} + 35 a + 10 + \left(21 a^{2} + 19 a + 34\right)\cdot 41 + \left(32 a^{2} + 9 a + 7\right)\cdot 41^{2} + \left(10 a^{2} + 16 a + 33\right)\cdot 41^{3} + \left(39 a^{2} + 20 a + 14\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 7 }$ | $=$ | \( 28 a^{2} + 4 a + 21 + \left(25 a^{2} + 36 a + 9\right)\cdot 41 + \left(5 a^{2} + 26 a + 17\right)\cdot 41^{2} + \left(11 a^{2} + 33\right)\cdot 41^{3} + \left(11 a^{2} + 8 a + 9\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 8 }$ | $=$ | \( 7 a^{2} + 3 a + 11 + \left(4 a^{2} + 2 a + 24\right)\cdot 41 + \left(23 a^{2} + 30 a + 14\right)\cdot 41^{2} + \left(20 a^{2} + 34 a + 32\right)\cdot 41^{3} + \left(37 a^{2} + 18 a + 7\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 9 }$ | $=$ | \( 31 a^{2} + 24 a + 27 + \left(12 a^{2} + 7 a + 2\right)\cdot 41 + \left(14 a^{2} + 15 a + 36\right)\cdot 41^{2} + \left(20 a^{2} + 23 a + 4\right)\cdot 41^{3} + \left(35 a^{2} + 9 a + 20\right)\cdot 41^{4} +O(41^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$9$ | $2$ | $(1,6)(2,3)(5,8)(7,9)$ | $0$ |
$2$ | $3$ | $(1,9,8)(2,4,3)(5,7,6)$ | $-1$ |
$2$ | $9$ | $(1,5,3,9,7,2,8,6,4)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$2$ | $9$ | $(1,3,7,8,4,5,9,2,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$2$ | $9$ | $(1,7,4,9,6,3,8,5,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.