Galois orbit of Artin representations 35.17e15_23e15_64879e15.70.1

• Label: 35.17e15_23e15_64879e15.70.1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 17^{15} \cdot 23^{15} \cdot 64879^{15}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$1159347343011657019888288788764768075653612272562741619862375047988448582213801308244211903468932991321422369849= 17^{15} \cdot 23^{15} \cdot 64879^{15} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 6 x^{5} + 4 x^{4} + 9 x^{3} - 4 x^{2} - 3 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 353 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 353 }$: $ x^{2} + 348 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 306 + 81\cdot 353 + 153\cdot 353^{2} + 289\cdot 353^{3} + 13\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 124 a + 13 + \left(4 a + 102\right)\cdot 353 + \left(266 a + 141\right)\cdot 353^{2} + \left(139 a + 292\right)\cdot 353^{3} + \left(10 a + 341\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 318 a + 298 + \left(37 a + 330\right)\cdot 353 + 301\cdot 353^{2} + \left(189 a + 227\right)\cdot 353^{3} + \left(22 a + 271\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 86 + 238\cdot 353 + 103\cdot 353^{2} + 195\cdot 353^{3} + 26\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 229 a + 280 + \left(348 a + 352\right)\cdot 353 + \left(86 a + 54\right)\cdot 353^{2} + \left(213 a + 19\right)\cdot 353^{3} + \left(342 a + 254\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 307 + 103\cdot 353 + 39\cdot 353^{2} + 274\cdot 353^{3} + 308\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 35 a + 123 + \left(315 a + 202\right)\cdot 353 + \left(352 a + 264\right)\cdot 353^{2} + \left(163 a + 113\right)\cdot 353^{3} + \left(330 a + 195\right)\cdot 353^{4} +O\left(353^{ 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
			$c1$
$1$	$1$	$()$	$35$
$21$	$2$	$(1,2)$	$5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$1$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$-1$
$630$	$4$	$(1,2,3,4)(5,6)$	$1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.