Artin representation 15.113e5_6761e5.42t412.1c1

• Label: 15.113e5_6761e5.42t412.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 113^{5} \cdot 6761^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$260283655280619872491891803193= 113^{5} \cdot 6761^{5} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} + x^{4} - 2 x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T412
Parity:	Even
Determinant:	1.113_6761.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $ x^{2} + 274 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 55 + 149\cdot 277 + 132\cdot 277^{2} + 244\cdot 277^{3} + 22\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 187 + 136\cdot 277 + 30\cdot 277^{2} + 274\cdot 277^{3} + 2\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 153 a + 49 + \left(9 a + 11\right)\cdot 277 + \left(254 a + 207\right)\cdot 277^{2} + \left(104 a + 234\right)\cdot 277^{3} + \left(223 a + 234\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 150 a + 27 + \left(207 a + 159\right)\cdot 277 + \left(182 a + 222\right)\cdot 277^{2} + \left(90 a + 101\right)\cdot 277^{3} + \left(221 a + 247\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 127 a + 200 + \left(69 a + 77\right)\cdot 277 + \left(94 a + 9\right)\cdot 277^{2} + \left(186 a + 191\right)\cdot 277^{3} + \left(55 a + 266\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 124 a + 231 + \left(267 a + 163\right)\cdot 277 + \left(22 a + 128\right)\cdot 277^{2} + \left(172 a + 18\right)\cdot 277^{3} + \left(53 a + 246\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 83 + 133\cdot 277 + 100\cdot 277^{2} + 43\cdot 277^{3} + 87\cdot 277^{4} +O\left(277^{ 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.