Artin representation 21.61e10_42899e10.84.1c1

• Label: 21.61e10_42899e10.84.1c1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 61^{10} \cdot 42899^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$15058095135110113805059217129191771009290371545949718904135503601= 61^{10} \cdot 42899^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 3 x^{5} + 4 x^{4} + x^{3} - 5 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 $\Q_{ 283 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 283 }$: $ x^{2} + 282 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 43 a + 260 + \left(202 a + 280\right)\cdot 283 + \left(238 a + 178\right)\cdot 283^{2} + \left(267 a + 124\right)\cdot 283^{3} + \left(54 a + 160\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 69 a + 113 + \left(181 a + 98\right)\cdot 283 + \left(241 a + 41\right)\cdot 283^{2} + \left(111 a + 43\right)\cdot 283^{3} + \left(117 a + 119\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 143 + 255\cdot 283 + 228\cdot 283^{2} + 67\cdot 283^{3} + 157\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 103 + 16\cdot 283 + 32\cdot 283^{2} + 5\cdot 283^{3} + 140\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 240 a + 20 + \left(80 a + 157\right)\cdot 283 + \left(44 a + 215\right)\cdot 283^{2} + \left(15 a + 153\right)\cdot 283^{3} + \left(228 a + 230\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 214 a + 182 + \left(101 a + 210\right)\cdot 283 + \left(41 a + 101\right)\cdot 283^{2} + \left(171 a + 196\right)\cdot 283^{3} + \left(165 a + 124\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 29 + 113\cdot 283 + 50\cdot 283^{2} + 258\cdot 283^{3} + 199\cdot 283^{4} +O\left(283^{ 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.