Artin representation 20.13e10_157e10_1367e10.70.1c1

• Label: 20.13e10_157e10_1367e10.70.1c1
• Dimension: 20
• Group: $S_7$
• Conductor: $ 13^{10} \cdot 157^{10} \cdot 1367^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$20$
Group:	$S_7$
Conductor:	$28583485949736448321604499971651223589810906282987259703935130049= 13^{10} \cdot 157^{10} \cdot 1367^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 2 + 27\cdot 29 + 28\cdot 29^{2} + 16\cdot 29^{3} + 24\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 10 a + 22 + \left(4 a + 19\right)\cdot 29 + \left(12 a + 5\right)\cdot 29^{2} + \left(10 a + 26\right)\cdot 29^{3} + \left(17 a + 10\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 6 a + 11 + \left(21 a + 6\right)\cdot 29 + \left(20 a + 15\right)\cdot 29^{2} + \left(23 a + 4\right)\cdot 29^{3} + \left(24 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 26 a + 21 + \left(17 a + 17\right)\cdot 29 + 18\cdot 29^{2} + \left(19 a + 18\right)\cdot 29^{3} + \left(25 a + 9\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 23 a + 12 + \left(7 a + 19\right)\cdot 29 + \left(8 a + 10\right)\cdot 29^{2} + \left(5 a + 15\right)\cdot 29^{3} + \left(4 a + 11\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 19 a + 14 + \left(24 a + 2\right)\cdot 29 + \left(16 a + 4\right)\cdot 29^{2} + \left(18 a + 8\right)\cdot 29^{3} + 11 a\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 3 a + 6 + \left(11 a + 23\right)\cdot 29 + \left(28 a + 3\right)\cdot 29^{2} + \left(9 a + 26\right)\cdot 29^{3} + \left(3 a + 2\right)\cdot 29^{4} +O\left(29^{ 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
$1$	$1$	$()$	$20$
$21$	$2$	$(1,2)$	$0$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$-4$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.