Artin representation 14.173e5_3821e5.30t565.1c1

• Label: 14.173e5_3821e5.30t565.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 173^{5} \cdot 3821^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$126216375357981850309278040393= 173^{5} \cdot 3821^{5} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} + 2 x^{3} + x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T565
Parity:	Even
Determinant:	1.173_3821.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 16 + 39\cdot 53 + 17\cdot 53^{2} + 46\cdot 53^{3} + 6\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 28 a + 39 + \left(38 a + 10\right)\cdot 53 + \left(27 a + 39\right)\cdot 53^{2} + \left(39 a + 38\right)\cdot 53^{3} + \left(10 a + 2\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 25 a + 45 + \left(14 a + 30\right)\cdot 53 + \left(25 a + 5\right)\cdot 53^{2} + \left(13 a + 10\right)\cdot 53^{3} + \left(42 a + 6\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 48 a + 49 + \left(34 a + 5\right)\cdot 53 + \left(32 a + 8\right)\cdot 53^{2} + \left(25 a + 48\right)\cdot 53^{3} + \left(44 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 5 a + 29 + \left(18 a + 44\right)\cdot 53 + \left(20 a + 50\right)\cdot 53^{2} + \left(27 a + 11\right)\cdot 53^{3} + \left(8 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 42 + 35\cdot 53 + 38\cdot 53^{2} + 31\cdot 53^{3} + 34\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 46 + 44\cdot 53 + 51\cdot 53^{2} + 24\cdot 53^{3} + 53^{4} +O\left(53^{ 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
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$-2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.