Artin representation 14.41e10_19433e10.42t413.1c1

• Label: 14.41e10_19433e10.42t413.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 41^{10} \cdot 19433^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$103094874408492845535079160737986296170946739175548537174049= 41^{10} \cdot 19433^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 2 x^{5} + 2 x^{4} + x^{3} - x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T413
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 499 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 499 }$: $ x^{2} + 493 x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 43 + 161\cdot 499 + 74\cdot 499^{2} + 499^{3} + 296\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 466 + 437\cdot 499 + 496\cdot 499^{2} + 488\cdot 499^{3} + 207\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 134 + 379\cdot 499 + 421\cdot 499^{2} + 15\cdot 499^{3} + 415\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 211 + 87\cdot 499 + 98\cdot 499^{2} + 353\cdot 499^{3} + 355\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 400 a + 481 + \left(324 a + 448\right)\cdot 499 + \left(456 a + 437\right)\cdot 499^{2} + \left(222 a + 270\right)\cdot 499^{3} + \left(449 a + 3\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 99 a + 386 + \left(174 a + 1\right)\cdot 499 + \left(42 a + 358\right)\cdot 499^{2} + \left(276 a + 153\right)\cdot 499^{3} + \left(49 a + 481\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 276 + 479\cdot 499 + 108\cdot 499^{2} + 213\cdot 499^{3} + 236\cdot 499^{4} +O\left(499^{ 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
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$-6$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-2$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.