Galois orbit of Artin representations 21.2975765603547883205205142226383560289457871113726968888545639351433.42t418.a

• Label: 21.297...433.42t418.a
• Dimension: $21$
• Group: $S_7$
• Conductor: $2.976\times 10^{66}$
• Indicator: $1$

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	\(297\!\cdots\!433\)\(\medspace = 1104217^{11} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 7.3.1104217.1
Galois orbit size:	$1$
Smallest permutation container:	42T418
Parity:	even
Projective image:	$S_7$
Projective field:	Galois closure of 7.3.1104217.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: \( x^{2} + 131x + 3 \)

Roots:

$r_{ 1 }$	$=$	\( 84 + 122\cdot 137 + 131\cdot 137^{2} + 74\cdot 137^{3} + 72\cdot 137^{4} +O(137^{5})\)
$r_{ 2 }$	$=$	\( 31 + 84\cdot 137 + 4\cdot 137^{2} + 117\cdot 137^{3} + 21\cdot 137^{4} +O(137^{5})\)
$r_{ 3 }$	$=$	\( 119 + 38\cdot 137 + 93\cdot 137^{2} + 31\cdot 137^{3} + 4\cdot 137^{4} +O(137^{5})\)
$r_{ 4 }$	$=$	\( 49 + 97\cdot 137 + 114\cdot 137^{2} + 26\cdot 137^{3} + 58\cdot 137^{4} +O(137^{5})\)
$r_{ 5 }$	$=$	\( 90 + 23\cdot 137 + 80\cdot 137^{2} + 60\cdot 137^{3} + 73\cdot 137^{4} +O(137^{5})\)
$r_{ 6 }$	$=$	\( 48 a + 82 + \left(107 a + 66\right)\cdot 137 + \left(117 a + 104\right)\cdot 137^{2} + \left(29 a + 87\right)\cdot 137^{3} + \left(17 a + 53\right)\cdot 137^{4} +O(137^{5})\)
$r_{ 7 }$	$=$	\( 89 a + 96 + \left(29 a + 114\right)\cdot 137 + \left(19 a + 18\right)\cdot 137^{2} + \left(107 a + 12\right)\cdot 137^{3} + \left(119 a + 127\right)\cdot 137^{4} +O(137^{5})\)

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation

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

$(1,2)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 7 }$	Character values
			$c1$
$1$	$1$	$()$	$21$
$21$	$2$	$(1,2)$	$-1$
$105$	$2$	$(1,2)(3,4)(5,6)$	$3$
$105$	$2$	$(1,2)(3,4)$	$1$
$70$	$3$	$(1,2,3)$	$-3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.