Artin representation 20.109e10_6389e10.70.1c1

• Label: 20.109e10_6389e10.70.1c1
• Dimension: 20
• Group: $S_7$
• Conductor: $ 109^{10} \cdot 6389^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$20$
Group:	$S_7$
Conductor:	$26828343653234231952951588152802844637789608782561070164001= 109^{10} \cdot 6389^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} + 2 x^{4} - 4 x^{3} + 4 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 $\Q_{ 71 }$ to precision 5.

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

Roots:

$r_{ 1 }$	$=$	$ 7 a + \left(13 a + 54\right)\cdot 71 + \left(44 a + 46\right)\cdot 71^{2} + \left(42 a + 13\right)\cdot 71^{3} + \left(16 a + 40\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 64 a + 14 + \left(57 a + 2\right)\cdot 71 + \left(26 a + 51\right)\cdot 71^{2} + \left(28 a + 54\right)\cdot 71^{3} + \left(54 a + 30\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 32 a + 38 + \left(49 a + 29\right)\cdot 71 + \left(52 a + 3\right)\cdot 71^{2} + \left(52 a + 25\right)\cdot 71^{3} + \left(56 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 33 a + 63 + \left(45 a + 41\right)\cdot 71 + \left(59 a + 50\right)\cdot 71^{2} + \left(30 a + 62\right)\cdot 71^{3} + \left(38 a + 1\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 10 + 31\cdot 71 + 19\cdot 71^{2} + 56\cdot 71^{3} + 7\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 38 a + 58 + \left(25 a + 28\right)\cdot 71 + \left(11 a + 53\right)\cdot 71^{2} + \left(40 a + 64\right)\cdot 71^{3} + \left(32 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 39 a + 31 + \left(21 a + 25\right)\cdot 71 + \left(18 a + 59\right)\cdot 71^{2} + \left(18 a + 6\right)\cdot 71^{3} + \left(14 a + 37\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
$1$	$1$	$()$	$20$
$21$	$2$	$(1,2)$	$0$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$-4$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.