Artin representation 20.89e10_365929e10.70.1c1

• Label: 20.89e10_365929e10.70.1c1
• Dimension: 20
• Group: $S_7$
• Conductor: $ 89^{10} \cdot 365929^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$20$
Group:	$S_7$
Conductor:	$1342357998274882740195075605250229122893802264603510088943494134090874924801= 89^{10} \cdot 365929^{10} $
Artin number field:	Splitting field of $f= x^{7} - 7 x^{5} - 2 x^{4} + 10 x^{3} + 3 x^{2} - 3 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 $\Q_{ 167 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 94 a + 118 + \left(120 a + 164\right)\cdot 167 + \left(106 a + 53\right)\cdot 167^{2} + \left(10 a + 81\right)\cdot 167^{3} + \left(101 a + 77\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 91 a + 49 + \left(3 a + 39\right)\cdot 167 + \left(103 a + 88\right)\cdot 167^{2} + \left(44 a + 26\right)\cdot 167^{3} + 102 a\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 79 + 4\cdot 167 + 141\cdot 167^{2} + 93\cdot 167^{3} + 147\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 43 + 40\cdot 167 + 154\cdot 167^{2} + 107\cdot 167^{3} + 80\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 27 + 109\cdot 167 + 2\cdot 167^{2} + 71\cdot 167^{3} + 136\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 73 a + 45 + \left(46 a + 24\right)\cdot 167 + \left(60 a + 40\right)\cdot 167^{2} + \left(156 a + 152\right)\cdot 167^{3} + 65 a\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 76 a + 140 + \left(163 a + 118\right)\cdot 167 + \left(63 a + 20\right)\cdot 167^{2} + \left(122 a + 135\right)\cdot 167^{3} + \left(64 a + 57\right)\cdot 167^{4} +O\left(167^{ 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.