Artin representation 15.313e5_1307e5.42t412.1c1

• Label: 15.313e5_1307e5.42t412.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 313^{5} \cdot 1307^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$11457757482495676836545566451= 313^{5} \cdot 1307^{5} $
Artin number field:	Splitting field of $f= x^{7} - x^{5} + 2 x^{3} - 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T412
Parity:	Odd
Determinant:	1.313_1307.2t1.1c1

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:

$r_{ 1 }$	$=$	$ 232 a + 227 + \left(27 a + 2\right)\cdot 353 + \left(298 a + 81\right)\cdot 353^{2} + \left(349 a + 121\right)\cdot 353^{3} + \left(152 a + 338\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 118 a + 151 + \left(107 a + 162\right)\cdot 353 + \left(302 a + 132\right)\cdot 353^{2} + \left(134 a + 105\right)\cdot 353^{3} + \left(84 a + 279\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 243 + 159\cdot 353 + 14\cdot 353^{2} + 247\cdot 353^{3} + 192\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 121 a + 328 + \left(325 a + 261\right)\cdot 353 + \left(54 a + 131\right)\cdot 353^{2} + \left(3 a + 160\right)\cdot 353^{3} + \left(200 a + 47\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 130 + 229\cdot 353 + 177\cdot 353^{2} + 315\cdot 353^{3} + 193\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 298 + 14\cdot 353 + 44\cdot 353^{2} + 338\cdot 353^{3} + 146\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 235 a + 35 + \left(245 a + 228\right)\cdot 353 + \left(50 a + 124\right)\cdot 353^{2} + \left(218 a + 124\right)\cdot 353^{3} + \left(268 a + 213\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 value
$1$	$1$	$()$	$15$
$21$	$2$	$(1,2)$	$5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-3$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.