Artin representation 2.1999.9t3.a.c

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

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

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 9 }$	Character value	Complex conjugation
$1$	$1$	$()$	$2$
$9$	$2$	$(1,6)(2,3)(5,8)(7,9)$	$0$	✓
$2$	$3$	$(1,9,8)(2,4,3)(5,7,6)$	$-1$
$2$	$9$	$(1,5,3,9,7,2,8,6,4)$	$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$
$2$	$9$	$(1,3,7,8,4,5,9,2,6)$	$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$
$2$	$9$	$(1,7,4,9,6,3,8,5,2)$	$\zeta_{9}^{5} + \zeta_{9}^{4}$