Artin representation 35.2e40_1787e20_11393e20.126.1c1

• Label: 35.2e40_1787e20_11393e20.126.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 2^{40} \cdot 1787^{20} \cdot 11393^{20}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$164608714851712408425439972790049716224466555819584932077978274357095613071205963906750204563256054529498408344034905956701093619769768465088006067253019672576= 2^{40} \cdot 1787^{20} \cdot 11393^{20} $
Artin number field:	Splitting field of $f= x^{7} - 7 x^{5} - x^{4} + 12 x^{3} + 2 x^{2} - 4 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 $\Q_{ 71 }$ to precision 5.

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

Roots:

$r_{ 1 }$	$=$	$ 27 + 38\cdot 71 + 53\cdot 71^{2} + 56\cdot 71^{3} + 55\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 55 a + 69 + \left(69 a + 50\right)\cdot 71 + \left(34 a + 62\right)\cdot 71^{2} + \left(22 a + 30\right)\cdot 71^{3} + \left(2 a + 69\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 51 a + 55 + \left(34 a + 38\right)\cdot 71 + \left(28 a + 5\right)\cdot 71^{2} + \left(22 a + 45\right)\cdot 71^{3} + \left(11 a + 29\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 21 a + 55 + 62\cdot 71 + \left(12 a + 23\right)\cdot 71^{2} + \left(44 a + 57\right)\cdot 71^{3} + \left(26 a + 54\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 20 a + 15 + \left(36 a + 57\right)\cdot 71 + \left(42 a + 27\right)\cdot 71^{2} + \left(48 a + 61\right)\cdot 71^{3} + \left(59 a + 29\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 16 a + 37 + \left(a + 64\right)\cdot 71 + \left(36 a + 62\right)\cdot 71^{2} + \left(48 a + 40\right)\cdot 71^{3} + \left(68 a + 51\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 50 a + 26 + \left(70 a + 42\right)\cdot 71 + \left(58 a + 47\right)\cdot 71^{2} + \left(26 a + 62\right)\cdot 71^{3} + \left(44 a + 63\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
$1$	$1$	$()$	$35$
$21$	$2$	$(1,2)$	$-5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-1$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$-1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.