Artin representation 14.11e5_89e5_389e5.30t565.1c1

• Label: 14.11e5_89e5_389e5.30t565.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 11^{5} \cdot 89^{5} \cdot 389^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$8010533954986717626675013151= 11^{5} \cdot 89^{5} \cdot 389^{5} $
Artin number field:	Splitting field of $f= x^{7} - x^{5} - x^{4} - x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T565
Parity:	Odd
Determinant:	1.11_89_389.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 22 a + 16 + \left(10 a + 30\right)\cdot 43 + \left(38 a + 23\right)\cdot 43^{2} + 16\cdot 43^{3} + \left(3 a + 24\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 32 a + 16 + \left(7 a + 36\right)\cdot 43 + \left(24 a + 36\right)\cdot 43^{2} + \left(33 a + 16\right)\cdot 43^{3} + \left(22 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 11 a + 5 + \left(35 a + 12\right)\cdot 43 + \left(18 a + 10\right)\cdot 43^{2} + \left(9 a + 26\right)\cdot 43^{3} + \left(20 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 5 + 8\cdot 43 + 14\cdot 43^{2} + 10\cdot 43^{3} +O\left(43^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 6 a + \left(4 a + 34\right)\cdot 43 + \left(25 a + 28\right)\cdot 43^{2} + \left(29 a + 37\right)\cdot 43^{3} + \left(23 a + 29\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 37 a + 6 + \left(38 a + 32\right)\cdot 43 + \left(17 a + 6\right)\cdot 43^{2} + \left(13 a + 42\right)\cdot 43^{3} + \left(19 a + 23\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 21 a + 38 + \left(32 a + 18\right)\cdot 43 + \left(4 a + 8\right)\cdot 43^{2} + \left(42 a + 22\right)\cdot 43^{3} + \left(39 a + 26\right)\cdot 43^{4} +O\left(43^{ 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
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$-2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.