Artin representation 18.11468902356161965397339320430592.36t1758.a.a

• Label: 18.114...592.36t1758.a.a
• Dimension: $18$
• Group: $S_4\wr C_2$
• Conductor: $1.147\times 10^{31}$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$18$
Group:	$S_4\wr C_2$
Conductor:	$114\!\cdots\!592$$\medspace = 2^{12} \cdot 3^{14} \cdot 7^{9} \cdot 29^{9}$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 8.2.57748079088.1
Galois orbit size:	$1$
Smallest permutation container:	36T1758
Parity:	odd
Determinant:	1.203.2t1.a.a
Projective image:	$S_4\wr C_2$
Projective stem field:	Galois closure of 8.2.57748079088.1

Defining polynomial

$f(x)$

$=$

$x^{8} - x^{7} + 5x^{6} - 4x^{5} + 10x^{4} + 10x^{3} - 4x^{2} + 4x + 4$ .

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $x^{2} + 58x + 2$

Roots:

$r_{ 1 }$	$=$	$57 a + 55 + \left(15 a + 49\right)\cdot 59 + \left(26 a + 14\right)\cdot 59^{2} + \left(4 a + 56\right)\cdot 59^{3} + \left(8 a + 43\right)\cdot 59^{4} + \left(13 a + 35\right)\cdot 59^{5} + \left(26 a + 10\right)\cdot 59^{6} + \left(49 a + 46\right)\cdot 59^{7} + \left(a + 39\right)\cdot 59^{8} + \left(22 a + 50\right)\cdot 59^{9} +O(59^{10})$
$r_{ 2 }$	$=$	$2 a + 53 + \left(43 a + 8\right)\cdot 59 + \left(32 a + 25\right)\cdot 59^{2} + \left(54 a + 34\right)\cdot 59^{3} + \left(50 a + 47\right)\cdot 59^{4} + \left(45 a + 40\right)\cdot 59^{5} + \left(32 a + 23\right)\cdot 59^{6} + \left(9 a + 10\right)\cdot 59^{7} + \left(57 a + 51\right)\cdot 59^{8} + \left(36 a + 11\right)\cdot 59^{9} +O(59^{10})$
$r_{ 3 }$	$=$	$5 a + 13 + \left(46 a + 1\right)\cdot 59 + \left(33 a + 1\right)\cdot 59^{2} + \left(25 a + 33\right)\cdot 59^{3} + \left(37 a + 53\right)\cdot 59^{4} + \left(25 a + 44\right)\cdot 59^{5} + \left(54 a + 40\right)\cdot 59^{6} + \left(45 a + 13\right)\cdot 59^{7} + \left(8 a + 50\right)\cdot 59^{8} + \left(36 a + 17\right)\cdot 59^{9} +O(59^{10})$
$r_{ 4 }$	$=$	$31 a + 20 + \left(2 a + 13\right)\cdot 59 + \left(52 a + 15\right)\cdot 59^{2} + \left(11 a + 29\right)\cdot 59^{3} + \left(50 a + 2\right)\cdot 59^{4} + \left(17 a + 57\right)\cdot 59^{5} + 21 a\cdot 59^{6} + \left(24 a + 3\right)\cdot 59^{7} + \left(13 a + 6\right)\cdot 59^{8} + \left(47 a + 49\right)\cdot 59^{9} +O(59^{10})$
$r_{ 5 }$	$=$	$28 a + 51 + \left(56 a + 43\right)\cdot 59 + \left(6 a + 5\right)\cdot 59^{2} + \left(47 a + 48\right)\cdot 59^{3} + \left(8 a + 40\right)\cdot 59^{4} + \left(41 a + 24\right)\cdot 59^{5} + \left(37 a + 4\right)\cdot 59^{6} + \left(34 a + 6\right)\cdot 59^{7} + \left(45 a + 54\right)\cdot 59^{8} + \left(11 a + 23\right)\cdot 59^{9} +O(59^{10})$
$r_{ 6 }$	$=$	$9 + 45\cdot 59 + 34\cdot 59^{2} + 37\cdot 59^{3} + 29\cdot 59^{4} + 20\cdot 59^{5} + 19\cdot 59^{6} + 2\cdot 59^{7} + 49\cdot 59^{8} + 57\cdot 59^{9} +O(59^{10})$
$r_{ 7 }$	$=$	$18 + 31\cdot 59 + 32\cdot 59^{2} + 31\cdot 59^{3} + 11\cdot 59^{4} + 38\cdot 59^{5} + 7\cdot 59^{6} + 31\cdot 59^{7} + 31\cdot 59^{8} + 38\cdot 59^{9} +O(59^{10})$
$r_{ 8 }$	$=$	$54 a + 18 + \left(12 a + 42\right)\cdot 59 + \left(25 a + 47\right)\cdot 59^{2} + \left(33 a + 24\right)\cdot 59^{3} + \left(21 a + 6\right)\cdot 59^{4} + \left(33 a + 33\right)\cdot 59^{5} + \left(4 a + 10\right)\cdot 59^{6} + \left(13 a + 5\right)\cdot 59^{7} + \left(50 a + 13\right)\cdot 59^{8} + \left(22 a + 45\right)\cdot 59^{9} +O(59^{10})$

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

Cycle notation

$(1,2,3,8)$

$(1,2)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character value	Complex conjugation
$1$	$1$	$()$	$18$
$6$	$2$	$(4,6)(5,7)$	$-6$
$9$	$2$	$(1,3)(2,8)(4,6)(5,7)$	$2$
$12$	$2$	$(1,2)$	$0$
$24$	$2$	$(1,4)(2,5)(3,6)(7,8)$	$0$
$36$	$2$	$(1,2)(4,5)$	$-2$
$36$	$2$	$(1,2)(4,6)(5,7)$	$0$	✓
$16$	$3$	$(1,3,8)$	$0$
$64$	$3$	$(1,3,8)(5,6,7)$	$0$
$12$	$4$	$(4,5,6,7)$	$0$
$36$	$4$	$(1,2,3,8)(4,5,6,7)$	$-2$
$36$	$4$	$(1,2,3,8)(4,6)(5,7)$	$0$
$72$	$4$	$(1,4,3,6)(2,5,8,7)$	$0$
$72$	$4$	$(1,2)(4,5,6,7)$	$2$
$144$	$4$	$(1,5,2,4)(3,6)(7,8)$	$0$
$48$	$6$	$(1,8,3)(4,6)(5,7)$	$0$
$96$	$6$	$(1,2)(5,7,6)$	$0$
$192$	$6$	$(1,5,3,6,8,7)(2,4)$	$0$
$144$	$8$	$(1,4,2,5,3,6,8,7)$	$0$
$96$	$12$	$(1,3,8)(4,5,6,7)$	$0$