Artin representation 20.132183947565219039133941155715525996025922875665780481.70.a.a

• Label: 20.132...481.70.a.a
• Dimension: $20$
• Group: $S_7$
• Conductor: $1.322\times 10^{53}$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$20$
Group:	$S_7$
Conductor:	$132\!\cdots\!481$$\medspace = 11^{10} \cdot 13^{12} \cdot 859^{10}$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 7.3.1596881.1
Galois orbit size:	$1$
Smallest permutation container:	70
Parity:	even
Determinant:	1.1.1t1.a.a
Projective image:	$S_7$
Projective stem field:	Galois closure of 7.3.1596881.1

Defining polynomial

$f(x)$

$=$

$x^{7} - 2x^{6} - 2x^{5} + 6x^{4} - x^{3} - 4x^{2} + 2x + 1$ .

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $x^{2} + 97x + 2$

Roots:

$r_{ 1 }$	$=$	$15 a + 56 + \left(84 a + 50\right)\cdot 101 + \left(47 a + 98\right)\cdot 101^{2} + \left(99 a + 100\right)\cdot 101^{3} + \left(56 a + 87\right)\cdot 101^{4} +O(101^{5})$
$r_{ 2 }$	$=$	$71 + 32\cdot 101 + 101^{2} + 34\cdot 101^{3} + 13\cdot 101^{4} +O(101^{5})$
$r_{ 3 }$	$=$	$34 a + 85 + \left(14 a + 43\right)\cdot 101 + \left(98 a + 28\right)\cdot 101^{2} + \left(55 a + 97\right)\cdot 101^{3} + \left(36 a + 99\right)\cdot 101^{4} +O(101^{5})$
$r_{ 4 }$	$=$	$26 + 5\cdot 101 + 88\cdot 101^{2} + 38\cdot 101^{3} + 9\cdot 101^{4} +O(101^{5})$
$r_{ 5 }$	$=$	$67 a + 19 + \left(86 a + 67\right)\cdot 101 + \left(2 a + 2\right)\cdot 101^{2} + \left(45 a + 21\right)\cdot 101^{3} + \left(64 a + 89\right)\cdot 101^{4} +O(101^{5})$
$r_{ 6 }$	$=$	$33 + 34\cdot 101 + 80\cdot 101^{2} + 64\cdot 101^{3} + 89\cdot 101^{4} +O(101^{5})$
$r_{ 7 }$	$=$	$86 a + 15 + \left(16 a + 69\right)\cdot 101 + \left(53 a + 3\right)\cdot 101^{2} + \left(a + 47\right)\cdot 101^{3} + \left(44 a + 14\right)\cdot 101^{4} +O(101^{5})$

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	Complex conjugation
$1$	$1$	$()$	$20$
$21$	$2$	$(1,2)$	$0$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$-4$	✓
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$