Artin representation 35.23e20_239e20_431e20.126.1c1

• Label: 35.23e20_239e20_431e20.126.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 23^{20} \cdot 239^{20} \cdot 431^{20}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$31049895475285366495114277243839735248438055901776217624431529186247392993227548396870393146919014699543281848616466503623924001= 23^{20} \cdot 239^{20} \cdot 431^{20} $
Artin number field:	Splitting field of $f= x^{7} - 4 x^{5} + 3 x^{3} - x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	126
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 4 + 16\cdot 37 + 3\cdot 37^{2} + 13\cdot 37^{3} + 19\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 19 a + 24 + \left(23 a + 13\right)\cdot 37 + \left(17 a + 29\right)\cdot 37^{2} + 35\cdot 37^{3} + \left(26 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 27 a + 16 + 3 a\cdot 37 + \left(26 a + 19\right)\cdot 37^{2} + 23 a\cdot 37^{3} + \left(11 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 32 a + 24 + \left(18 a + 35\right)\cdot 37 + \left(16 a + 18\right)\cdot 37^{2} + \left(24 a + 19\right)\cdot 37^{3} + \left(14 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 5 a + 4 + \left(18 a + 5\right)\cdot 37 + \left(20 a + 29\right)\cdot 37^{2} + \left(12 a + 26\right)\cdot 37^{3} + \left(22 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 18 a + 26 + \left(13 a + 14\right)\cdot 37 + \left(19 a + 2\right)\cdot 37^{2} + \left(36 a + 20\right)\cdot 37^{3} + \left(10 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 10 a + 13 + \left(33 a + 25\right)\cdot 37 + \left(10 a + 8\right)\cdot 37^{2} + \left(13 a + 32\right)\cdot 37^{3} + \left(25 a + 16\right)\cdot 37^{4} +O\left(37^{ 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
$1$	$1$	$()$	$35$
$21$	$2$	$(1,2)$	$-5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-1$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$-1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.