Artin representation 35.107e15_21557e15.70.1c1

• Label: 35.107e15_21557e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 107^{15} \cdot 21557^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$278343760981136042120801246774474576193180676510700270347019900817402542277132910496604340798999= 107^{15} \cdot 21557^{15} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{5} - x^{4} + x^{3} + 3 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Odd
Determinant:	1.107_21557.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 48 a + 16 + \left(4 a + 4\right)\cdot 53 + \left(46 a + 6\right)\cdot 53^{2} + \left(38 a + 39\right)\cdot 53^{3} + \left(24 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 41 a + 14 + \left(51 a + 30\right)\cdot 53 + \left(17 a + 47\right)\cdot 53^{2} + \left(31 a + 31\right)\cdot 53^{3} + \left(4 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 33 a + 13 + \left(41 a + 24\right)\cdot 53 + \left(38 a + 5\right)\cdot 53^{2} + \left(50 a + 33\right)\cdot 53^{3} + \left(14 a + 8\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 5 a + 49 + \left(48 a + 28\right)\cdot 53 + \left(6 a + 26\right)\cdot 53^{2} + \left(14 a + 42\right)\cdot 53^{3} + \left(28 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 20 a + 39 + \left(11 a + 51\right)\cdot 53 + \left(14 a + 12\right)\cdot 53^{2} + \left(2 a + 38\right)\cdot 53^{3} + \left(38 a + 17\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 9 + 35\cdot 53 + 45\cdot 53^{2} + 46\cdot 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 12 a + 19 + \left(a + 37\right)\cdot 53 + \left(35 a + 14\right)\cdot 53^{2} + \left(21 a + 33\right)\cdot 53^{3} + \left(48 a + 47\right)\cdot 53^{4} +O\left(53^{ 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.