Galois orbit of Artin representations 21.2e16_29e11_47e11_14563e11.42t418.1

• Label: 21.2e16_29e11_47e11_14563e11.42t418.1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 2^{16} \cdot 29^{11} \cdot 47^{11} \cdot 14563^{11}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$12350760194986624716747310006119513115571299032855591720951766409772480921787339177984= 2^{16} \cdot 29^{11} \cdot 47^{11} \cdot 14563^{11} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{6} - 3 x^{5} + 13 x^{4} - x^{3} - 12 x^{2} + 2 x + 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T418
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$: $ x^{2} + 192 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 23 a + 18 + \left(13 a + 185\right)\cdot 197 + \left(157 a + 84\right)\cdot 197^{2} + \left(72 a + 146\right)\cdot 197^{3} + \left(125 a + 111\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 193 a + 17 + \left(152 a + 10\right)\cdot 197 + \left(114 a + 144\right)\cdot 197^{2} + \left(68 a + 131\right)\cdot 197^{3} + \left(182 a + 44\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 153 + 2\cdot 197 + 71\cdot 197^{2} + 5\cdot 197^{3} + 137\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 76 + 180\cdot 197 + 132\cdot 197^{2} + 177\cdot 197^{3} + 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 174 a + 133 + \left(183 a + 30\right)\cdot 197 + \left(39 a + 69\right)\cdot 197^{2} + \left(124 a + 156\right)\cdot 197^{3} + \left(71 a + 74\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 3 + 191\cdot 197 + 114\cdot 197^{2} + 7\cdot 197^{3} + 121\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 4 a + 194 + \left(44 a + 187\right)\cdot 197 + \left(82 a + 170\right)\cdot 197^{2} + \left(128 a + 162\right)\cdot 197^{3} + \left(14 a + 99\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$

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.