Artin representation 6.23e5_39551e5.14t46.1c1

• Label: 6.23e5_39551e5.14t46.1c1
• Dimension: 6
• Group: $S_7$
• Conductor: $ 23^{5} \cdot 39551^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$6$
Group:	$S_7$
Conductor:	$622911749984084961078380339593= 23^{5} \cdot 39551^{5} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + 3 x^{5} - 4 x^{4} + 2 x^{3} - x^{2} - x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	14T46
Parity:	Even
Determinant:	1.23_39551.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 4 + 24\cdot 79 + 59\cdot 79^{2} + 14\cdot 79^{3} + 75\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 25 a + 13 + \left(41 a + 63\right)\cdot 79 + \left(38 a + 63\right)\cdot 79^{2} + \left(62 a + 22\right)\cdot 79^{3} + \left(21 a + 76\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 37 + 7\cdot 79 + 29\cdot 79^{2} + 22\cdot 79^{3} + 78\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 54 a + 38 + 37 a\cdot 79 + \left(40 a + 61\right)\cdot 79^{2} + \left(16 a + 46\right)\cdot 79^{3} + \left(57 a + 35\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 39 + 18\cdot 79^{2} + 75\cdot 79^{3} + 30\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 51 a + 68 + \left(47 a + 32\right)\cdot 79 + \left(66 a + 72\right)\cdot 79^{2} + 40 a\cdot 79^{3} + \left(74 a + 72\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 28 a + 40 + \left(31 a + 29\right)\cdot 79 + \left(12 a + 12\right)\cdot 79^{2} + \left(38 a + 54\right)\cdot 79^{3} + \left(4 a + 26\right)\cdot 79^{4} +O\left(79^{ 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
$1$	$1$	$()$	$6$
$21$	$2$	$(1,2)$	$-4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$-2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.