Galois orbit of Artin representations 9.518705353233.18t272.a

• Label: 9.518705353233.18t272.a
• Dimension: $9$
• Group: $S_4\wr C_2$
• Conductor: $518705353233$
• Indicator: $1$

Basic invariants

Dimension:	$9$
Group:	$S_4\wr C_2$
Conductor:	\(518705353233\)\(\medspace = 3^{7} \cdot 619^{3} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 8.6.321078613651227.1
Galois orbit size:	$1$
Smallest permutation container:	18T272
Parity:	even
Projective image:	$S_4\wr C_2$
Projective field:	Galois closure of 8.6.321078613651227.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)

Roots:

$r_{ 1 }$	$=$	\( 25 a + 28 + \left(8 a + 27\right)\cdot 29 + 25\cdot 29^{2} + \left(27 a + 14\right)\cdot 29^{3} + \left(19 a + 22\right)\cdot 29^{4} + \left(21 a + 2\right)\cdot 29^{5} + \left(25 a + 14\right)\cdot 29^{6} + 7 a\cdot 29^{7} + \left(4 a + 15\right)\cdot 29^{8} + \left(11 a + 10\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 2 }$	$=$	\( 14 + 10\cdot 29 + 11\cdot 29^{2} + 4\cdot 29^{3} + 8\cdot 29^{4} + 15\cdot 29^{5} + 15\cdot 29^{6} + 25\cdot 29^{7} + 21\cdot 29^{8} + 8\cdot 29^{9} +O(29^{10})\)
$r_{ 3 }$	$=$	\( 28 a + 7 a\cdot 29 + \left(22 a + 13\right)\cdot 29^{2} + 17 a\cdot 29^{3} + \left(28 a + 9\right)\cdot 29^{4} + \left(3 a + 15\right)\cdot 29^{5} + \left(27 a + 11\right)\cdot 29^{6} + \left(28 a + 6\right)\cdot 29^{7} + \left(19 a + 15\right)\cdot 29^{8} + \left(26 a + 8\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 4 }$	$=$	\( a + 24 + \left(21 a + 11\right)\cdot 29 + 6 a\cdot 29^{2} + \left(11 a + 9\right)\cdot 29^{3} + 18\cdot 29^{4} + \left(25 a + 6\right)\cdot 29^{5} + \left(a + 27\right)\cdot 29^{6} + 7\cdot 29^{7} + \left(9 a + 28\right)\cdot 29^{8} + \left(2 a + 5\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 5 }$	$=$	\( 26 a + 19 + \left(18 a + 2\right)\cdot 29 + \left(19 a + 15\right)\cdot 29^{2} + \left(18 a + 18\right)\cdot 29^{3} + \left(5 a + 4\right)\cdot 29^{4} + \left(7 a + 21\right)\cdot 29^{5} + \left(6 a + 2\right)\cdot 29^{6} + \left(24 a + 10\right)\cdot 29^{7} + \left(8 a + 17\right)\cdot 29^{8} + \left(a + 10\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 6 }$	$=$	\( 23 + 2\cdot 29 + 24\cdot 29^{2} + 2\cdot 29^{4} + 28\cdot 29^{5} + 12\cdot 29^{6} + 13\cdot 29^{7} + 10\cdot 29^{8} + 29^{9} +O(29^{10})\)
$r_{ 7 }$	$=$	\( 4 a + 8 + \left(20 a + 18\right)\cdot 29 + \left(28 a + 18\right)\cdot 29^{2} + \left(a + 4\right)\cdot 29^{3} + \left(9 a + 8\right)\cdot 29^{4} + \left(7 a + 4\right)\cdot 29^{5} + \left(3 a + 5\right)\cdot 29^{6} + \left(21 a + 14\right)\cdot 29^{7} + \left(24 a + 28\right)\cdot 29^{8} + \left(17 a + 3\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 8 }$	$=$	\( 3 a + 4 + \left(10 a + 13\right)\cdot 29 + \left(9 a + 7\right)\cdot 29^{2} + \left(10 a + 5\right)\cdot 29^{3} + \left(23 a + 14\right)\cdot 29^{4} + \left(21 a + 22\right)\cdot 29^{5} + \left(22 a + 26\right)\cdot 29^{6} + \left(4 a + 8\right)\cdot 29^{7} + \left(20 a + 8\right)\cdot 29^{8} + \left(27 a + 8\right)\cdot 29^{9} +O(29^{10})\)

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(1,3,4,7)$

$(1,3)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$9$
$6$	$2$	$(1,4)(3,7)$	$-3$
$9$	$2$	$(1,4)(2,6)(3,7)(5,8)$	$1$
$12$	$2$	$(1,3)$	$-3$
$24$	$2$	$(1,2)(3,5)(4,6)(7,8)$	$3$
$36$	$2$	$(1,3)(2,5)$	$1$
$36$	$2$	$(1,3)(2,6)(5,8)$	$1$
$16$	$3$	$(3,4,7)$	$0$
$64$	$3$	$(3,4,7)(5,6,8)$	$0$
$12$	$4$	$(1,3,4,7)$	$3$
$36$	$4$	$(1,3,4,7)(2,5,6,8)$	$1$
$36$	$4$	$(1,4)(2,5,6,8)(3,7)$	$-1$
$72$	$4$	$(1,6,4,2)(3,8,7,5)$	$-1$
$72$	$4$	$(1,3)(2,5,6,8)$	$-1$
$144$	$4$	$(1,5,3,2)(4,6)(7,8)$	$-1$
$48$	$6$	$(2,6)(3,7,4)(5,8)$	$0$
$96$	$6$	$(1,3)(2,6,8)$	$0$
$192$	$6$	$(1,2)(3,6,4,8,7,5)$	$0$
$144$	$8$	$(1,5,3,6,4,8,7,2)$	$1$
$96$	$12$	$(2,5,6,8)(3,4,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.