Artin representation 6.3e2_114089e5.14t46.1c1

• Label: 6.3e2_114089e5.14t46.1c1
• Dimension: 6
• Group: $S_7$
• Conductor: $ 3^{2} \cdot 114089^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$6$
Group:	$S_7$
Conductor:	$173964797962864019112865041= 3^{2} \cdot 114089^{5} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} - x^{4} + x^{3} + x^{2} + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	14T46
Parity:	Even
Determinant:	1.114089.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 499 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 499 }$: $ x^{2} + 493 x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 22 a + 40 + \left(438 a + 386\right)\cdot 499 + \left(151 a + 357\right)\cdot 499^{2} + \left(445 a + 202\right)\cdot 499^{3} + \left(269 a + 400\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 477 a + 172 + \left(60 a + 497\right)\cdot 499 + \left(347 a + 331\right)\cdot 499^{2} + \left(53 a + 227\right)\cdot 499^{3} + \left(229 a + 77\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 116 + 307\cdot 499 + 156\cdot 499^{2} + 237\cdot 499^{3} + 223\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 165 a + 449 + \left(240 a + 348\right)\cdot 499 + \left(343 a + 343\right)\cdot 499^{2} + \left(396 a + 347\right)\cdot 499^{3} + \left(39 a + 104\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 334 a + 441 + \left(258 a + 128\right)\cdot 499 + \left(155 a + 168\right)\cdot 499^{2} + \left(102 a + 388\right)\cdot 499^{3} + \left(459 a + 445\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 267 + 81\cdot 499 + 292\cdot 499^{2} + 471\cdot 499^{3} + 450\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 13 + 246\cdot 499 + 345\cdot 499^{2} + 120\cdot 499^{3} + 293\cdot 499^{4} +O\left(499^{ 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
$1$	$1$	$()$	$6$
$21$	$2$	$(1,2)$	$-4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$-2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.