Artin representation 35.31e15_25447e15.70.1c1

• Label: 35.31e15_25447e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 31^{15} \cdot 25447^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$28508492614653806795875851456329273740583676620858122101344290603152657886181201620677193= 31^{15} \cdot 25447^{15} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} + 3 x^{4} - 2 x^{3} + 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Even
Determinant:	1.31_25447.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 2 + 41\cdot 67 + 23\cdot 67^{2} + 62\cdot 67^{3} + 65\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 3 a + 50 + \left(36 a + 8\right)\cdot 67 + \left(4 a + 44\right)\cdot 67^{2} + \left(3 a + 4\right)\cdot 67^{3} + \left(19 a + 36\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 26 a + 38 + \left(58 a + 44\right)\cdot 67 + \left(2 a + 7\right)\cdot 67^{2} + \left(47 a + 52\right)\cdot 67^{3} + \left(48 a + 10\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 64 a + 62 + \left(30 a + 15\right)\cdot 67 + \left(62 a + 26\right)\cdot 67^{2} + \left(63 a + 12\right)\cdot 67^{3} + \left(47 a + 42\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 40 a + 8 + \left(49 a + 41\right)\cdot 67 + \left(16 a + 60\right)\cdot 67^{2} + \left(40 a + 44\right)\cdot 67^{3} + \left(38 a + 20\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 41 a + 8 + \left(8 a + 51\right)\cdot 67 + \left(64 a + 27\right)\cdot 67^{2} + \left(19 a + 36\right)\cdot 67^{3} + \left(18 a + 24\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 27 a + 34 + \left(17 a + 65\right)\cdot 67 + \left(50 a + 10\right)\cdot 67^{2} + \left(26 a + 55\right)\cdot 67^{3} + 28 a\cdot 67^{4} +O\left(67^{ 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.