Artin representation 14.3e6_37e5_2381e5.30t565.1c1

• Label: 14.3e6_37e5_2381e5.30t565.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 3^{6} \cdot 37^{5} \cdot 2381^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$3868415597281715332824607353= 3^{6} \cdot 37^{5} \cdot 2381^{5} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{5} + x^{4} - 2 x^{3} + 2 x^{2} - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T565
Parity:	Even
Determinant:	1.37_2381.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 2 + 21\cdot 71 + 57\cdot 71^{2} + 50\cdot 71^{3} + 52\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 61 a + 55 + \left(31 a + 11\right)\cdot 71 + \left(70 a + 54\right)\cdot 71^{2} + \left(6 a + 64\right)\cdot 71^{3} + \left(51 a + 3\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 38 + 71 + 38\cdot 71^{2} + 23\cdot 71^{3} + 46\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 10 a + 35 + \left(39 a + 14\right)\cdot 71 + 21\cdot 71^{2} + \left(64 a + 8\right)\cdot 71^{3} + \left(19 a + 28\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 32 a + 33 + \left(17 a + 12\right)\cdot 71 + \left(25 a + 43\right)\cdot 71^{2} + \left(16 a + 61\right)\cdot 71^{3} + \left(39 a + 10\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 26 + 65\cdot 71 + 64\cdot 71^{2} + 5\cdot 71^{3} + 69\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 39 a + 26 + \left(53 a + 15\right)\cdot 71 + \left(45 a + 5\right)\cdot 71^{2} + \left(54 a + 69\right)\cdot 71^{3} + \left(31 a + 1\right)\cdot 71^{4} +O\left(71^{ 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.