Artin representation 21.43e10_107e10_233e10.84.1c1

• Label: 21.43e10_107e10_233e10.84.1c1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 43^{10} \cdot 107^{10} \cdot 233^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$2004848421334875710954164439986393698679641661589828401424449= 43^{10} \cdot 107^{10} \cdot 233^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{5} - x^{4} - x^{3} - x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	84
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 39 a + 65 + \left(122 a + 66\right)\cdot 127 + \left(57 a + 110\right)\cdot 127^{2} + \left(29 a + 34\right)\cdot 127^{3} + \left(95 a + 106\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 92 + 64\cdot 127 + 32\cdot 127^{2} + 23\cdot 127^{3} + 25\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 12 + 69\cdot 127 + 125\cdot 127^{2} + 63\cdot 127^{3} + 12\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 94 a + 42 + \left(80 a + 11\right)\cdot 127 + \left(63 a + 16\right)\cdot 127^{2} + \left(7 a + 76\right)\cdot 127^{3} + \left(52 a + 104\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 33 a + 9 + \left(46 a + 125\right)\cdot 127 + \left(63 a + 125\right)\cdot 127^{2} + \left(119 a + 19\right)\cdot 127^{3} + \left(74 a + 22\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 57 + 21\cdot 127 + 51\cdot 127^{2} + 29\cdot 127^{3} + 65\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 88 a + 104 + \left(4 a + 22\right)\cdot 127 + \left(69 a + 46\right)\cdot 127^{2} + \left(97 a + 6\right)\cdot 127^{3} + \left(31 a + 45\right)\cdot 127^{4} +O\left(127^{ 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 value
$1$	$1$	$()$	$21$
$21$	$2$	$(1,2)$	$1$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-3$
$105$	$2$	$(1,2)(3,4)$	$1$
$70$	$3$	$(1,2,3)$	$-3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$-1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.