Galois orbit of Artin representations 9.2e6_3e6_17e6_23e6.20t145.1

• Label: 9.2e6_3e6_17e6_23e6.20t145.1
• Dimension: 9
• Group: $S_6$
• Conductor: $ 2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 23^{6}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$9$
Group:	$S_6$
Conductor:	$166712454894136965696= 2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 23^{6} $
Artin number field:	Splitting field of $f= x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 2 x^{2} + 3 x - 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	20T145
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 401 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 401 }$: $ x^{2} + 396 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 195 a + 103 + \left(352 a + 27\right)\cdot 401 + \left(127 a + 120\right)\cdot 401^{2} + \left(73 a + 211\right)\cdot 401^{3} + \left(331 a + 187\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 349 + 218\cdot 401 + 191\cdot 401^{2} + 127\cdot 401^{3} + 374\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 206 a + 276 + \left(48 a + 391\right)\cdot 401 + \left(273 a + 5\right)\cdot 401^{2} + \left(327 a + 49\right)\cdot 401^{3} + \left(69 a + 166\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 371 + 317\cdot 401 + 306\cdot 401^{2} + 34\cdot 401^{3} + 14\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 62 a + 98 + \left(377 a + 214\right)\cdot 401 + \left(139 a + 328\right)\cdot 401^{2} + \left(163 a + 51\right)\cdot 401^{3} + \left(256 a + 72\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 339 a + 7 + \left(23 a + 33\right)\cdot 401 + \left(261 a + 250\right)\cdot 401^{2} + \left(237 a + 327\right)\cdot 401^{3} + \left(144 a + 388\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation

$(1,2)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$9$
$15$	$2$	$(1,2)(3,4)(5,6)$	$-3$
$15$	$2$	$(1,2)$	$-3$
$45$	$2$	$(1,2)(3,4)$	$1$
$40$	$3$	$(1,2,3)(4,5,6)$	$0$
$40$	$3$	$(1,2,3)$	$0$
$90$	$4$	$(1,2,3,4)(5,6)$	$1$
$90$	$4$	$(1,2,3,4)$	$1$
$144$	$5$	$(1,2,3,4,5)$	$-1$
$120$	$6$	$(1,2,3,4,5,6)$	$0$
$120$	$6$	$(1,2,3)(4,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.