Properties

Label 2.1999.9t3.a.a
Dimension $2$
Group $D_{9}$
Conductor $1999$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{9}$
Conductor: \(1999\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 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: 9.1.15968023992001.1

Defining polynomial

$f(x)$$=$\(x^{9} - 3 x^{8} - 2 x^{7} + 3 x^{6} + 19 x^{5} + 39 x^{4} + 85 x^{3} + 66 x^{2} + 108 x + 27\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \(x^{3} + x + 35\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a^{2} + 14 a + 22 + \left(24 a^{2} + 31 a + 37\right)\cdot 41 + \left(3 a^{2} + 36 a + 28\right)\cdot 41^{2} + \left(23 a + 18\right)\cdot 41^{3} + \left(9 a^{2} + 12 a + 2\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 26 a^{2} + 32 a + 37 + \left(3 a^{2} + 40 a + 15\right)\cdot 41 + \left(11 a^{2} + 14 a + 8\right)\cdot 41^{2} + \left(26 a^{2} + 32 a\right)\cdot 41^{3} + \left(20 a^{2} + 22 a + 15\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 a^{2} + 15 a + 16 + \left(24 a^{2} + 34 a + 2\right)\cdot 41 + \left(12 a^{2} + 30 a + 23\right)\cdot 41^{2} + \left(2 a^{2} + 37 a + 11\right)\cdot 41^{3} + \left(15 a^{2} + 13 a + 11\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 35 a + 6 + \left(13 a^{2} + 6 a + 22\right)\cdot 41 + \left(17 a^{2} + 36 a + 12\right)\cdot 41^{2} + \left(12 a^{2} + 11 a + 18\right)\cdot 41^{3} + \left(5 a^{2} + 4 a + 18\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 22 a^{2} + 2 a + 17 + \left(34 a^{2} + 26 a + 15\right)\cdot 41 + \left(2 a^{2} + 4 a + 15\right)\cdot 41^{2} + \left(19 a^{2} + 24 a + 11\right)\cdot 41^{3} + \left(31 a^{2} + 12 a + 23\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 32 a^{2} + 35 a + 10 + \left(21 a^{2} + 19 a + 34\right)\cdot 41 + \left(32 a^{2} + 9 a + 7\right)\cdot 41^{2} + \left(10 a^{2} + 16 a + 33\right)\cdot 41^{3} + \left(39 a^{2} + 20 a + 14\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 28 a^{2} + 4 a + 21 + \left(25 a^{2} + 36 a + 9\right)\cdot 41 + \left(5 a^{2} + 26 a + 17\right)\cdot 41^{2} + \left(11 a^{2} + 33\right)\cdot 41^{3} + \left(11 a^{2} + 8 a + 9\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 7 a^{2} + 3 a + 11 + \left(4 a^{2} + 2 a + 24\right)\cdot 41 + \left(23 a^{2} + 30 a + 14\right)\cdot 41^{2} + \left(20 a^{2} + 34 a + 32\right)\cdot 41^{3} + \left(37 a^{2} + 18 a + 7\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 31 a^{2} + 24 a + 27 + \left(12 a^{2} + 7 a + 2\right)\cdot 41 + \left(14 a^{2} + 15 a + 36\right)\cdot 41^{2} + \left(20 a^{2} + 23 a + 4\right)\cdot 41^{3} + \left(35 a^{2} + 9 a + 20\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(1,6)(2,3)(5,8)(7,9)$
$(1,5,3,9,7,2,8,6,4)$
$(1,9,8)(2,4,3)(5,7,6)$

Character values on conjugacy classes

SizeOrderAction 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}^{4} + \zeta_{9}^{2} - \zeta_{9}$
$2$$9$$(1,3,7,8,4,5,9,2,6)$$\zeta_{9}^{5} + \zeta_{9}^{4}$
$2$$9$$(1,7,4,9,6,3,8,5,2)$$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$

The blue line marks the conjugacy class containing complex conjugation.