Artin representation 35.89e15_67231e15.70.1c1

• Label: 35.89e15_67231e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 89^{15} \cdot 67231^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$451225531583577276455455583464836976639627340261042414185935261597586474811862426789951521681630705399= 89^{15} \cdot 67231^{15} $
Artin number field:	Splitting field of $f= x^{7} - x^{5} - x^{4} - 3 x^{3} + 2 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Odd
Determinant:	1.89_67231.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: $ x^{2} + 145 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 9 a + 105 + \left(97 a + 6\right)\cdot 149 + \left(147 a + 94\right)\cdot 149^{2} + \left(110 a + 44\right)\cdot 149^{3} + \left(14 a + 8\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 104 a + 8 + \left(41 a + 33\right)\cdot 149 + \left(25 a + 78\right)\cdot 149^{2} + \left(98 a + 133\right)\cdot 149^{3} + \left(97 a + 101\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 95 + 30\cdot 149 + 148\cdot 149^{2} + 81\cdot 149^{3} + 57\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 45 a + 126 + \left(107 a + 95\right)\cdot 149 + \left(123 a + 137\right)\cdot 149^{2} + \left(50 a + 53\right)\cdot 149^{3} + \left(51 a + 96\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 74 + 133\cdot 149 + 118\cdot 149^{2} + 12\cdot 149^{3} + 43\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 140 a + 141 + \left(51 a + 87\right)\cdot 149 + \left(a + 140\right)\cdot 149^{2} + \left(38 a + 42\right)\cdot 149^{3} + \left(134 a + 105\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 47 + 59\cdot 149 + 27\cdot 149^{2} + 77\cdot 149^{3} + 34\cdot 149^{4} +O\left(149^{ 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.