Artin representation 15.263e10_4271e10.42t411.1c1

• Label: 15.263e10_4271e10.42t411.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 263^{10} \cdot 4271^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$3197813997378074485482319022511192751480727538874362307173649= 263^{10} \cdot 4271^{10} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{5} - 2 x^{3} + 2 x^{2} - x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T411
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$: $ x^{2} + 177 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 126 a + 70 + \left(128 a + 126\right)\cdot 181 + \left(12 a + 77\right)\cdot 181^{2} + \left(177 a + 73\right)\cdot 181^{3} + \left(130 a + 150\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 172 + 110\cdot 181 + 37\cdot 181^{2} + 135\cdot 181^{3} + 23\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 61 a + 92 + \left(7 a + 51\right)\cdot 181 + \left(29 a + 92\right)\cdot 181^{2} + \left(111 a + 118\right)\cdot 181^{3} + \left(90 a + 114\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 152 + 135\cdot 181 + 154\cdot 181^{2} + 113\cdot 181^{3} + 149\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 120 a + 155 + \left(173 a + 19\right)\cdot 181 + \left(151 a + 20\right)\cdot 181^{2} + \left(69 a + 172\right)\cdot 181^{3} + \left(90 a + 3\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 55 a + 31 + \left(52 a + 153\right)\cdot 181 + \left(168 a + 180\right)\cdot 181^{2} + \left(3 a + 44\right)\cdot 181^{3} + \left(50 a + 135\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 54 + 126\cdot 181 + 160\cdot 181^{2} + 65\cdot 181^{3} + 146\cdot 181^{4} +O\left(181^{ 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.