Artin representation 15.41e5_25537e5.42t412.1c1

• Label: 15.41e5_25537e5.42t412.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 41^{5} \cdot 25537^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$1258255003371595820655403854857= 41^{5} \cdot 25537^{5} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{5} - 2 x^{4} + x^{3} + x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T412
Parity:	Even
Determinant:	1.41_25537.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: $ x^{2} + 190 x + 19 $

Roots:

$r_{ 1 }$	$=$	$ 9 + 122\cdot 191 + 13\cdot 191^{2} + 80\cdot 191^{3} + 123\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 164 + 101\cdot 191 + 54\cdot 191^{2} + 4\cdot 191^{3} + 182\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 151 + 11\cdot 191 + 81\cdot 191^{2} + 74\cdot 191^{3} + 118\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 68 a + 76 + \left(94 a + 116\right)\cdot 191 + \left(183 a + 115\right)\cdot 191^{2} + \left(163 a + 18\right)\cdot 191^{3} + \left(126 a + 72\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 179 + 102\cdot 191 + 171\cdot 191^{2} + 138\cdot 191^{3} + 69\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 123 a + 144 + \left(96 a + 142\right)\cdot 191 + \left(7 a + 13\right)\cdot 191^{2} + \left(27 a + 190\right)\cdot 191^{3} + \left(64 a + 34\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 41 + 166\cdot 191 + 122\cdot 191^{2} + 66\cdot 191^{3} + 163\cdot 191^{4} +O\left(191^{ 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.