Galois orbit of Artin representations 21.7398903246368894437623726246710239803062398398196631330449.84.a

• Label: 21.739...449.84.a
• Dimension: $21$
• Group: $S_7$
• Conductor: $7.399\times 10^{57}$
• Indicator: $1$

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	\(739\!\cdots\!449\)\(\medspace = 71^{10} \cdot 8623^{10} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 7.3.612233.1
Galois orbit size:	$1$
Smallest permutation container:	84
Parity:	even
Projective image:	$S_7$
Projective field:	Galois closure of 7.3.612233.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \)

Roots:

$r_{ 1 }$	$=$	\( 25 a + 21 + \left(6 a + 20\right)\cdot 31 + \left(22 a + 19\right)\cdot 31^{2} + \left(6 a + 13\right)\cdot 31^{3} + \left(4 a + 24\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$	$=$	\( 27 + 5\cdot 31 + 12\cdot 31^{2} + 12\cdot 31^{3} + 14\cdot 31^{4} +O(31^{5})\)
$r_{ 3 }$	$=$	\( 28 a + 21 + 25\cdot 31 + \left(10 a + 16\right)\cdot 31^{2} + \left(16 a + 10\right)\cdot 31^{3} + \left(5 a + 24\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 4 }$	$=$	\( 10 + 28\cdot 31 + 17\cdot 31^{2} + 7\cdot 31^{3} + 19\cdot 31^{4} +O(31^{5})\)
$r_{ 5 }$	$=$	\( 3 a + 15 + \left(30 a + 30\right)\cdot 31 + \left(20 a + 4\right)\cdot 31^{2} + \left(14 a + 2\right)\cdot 31^{3} + \left(25 a + 19\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 6 }$	$=$	\( 22 + 3\cdot 31 + 26\cdot 31^{2} + 10\cdot 31^{3} + 27\cdot 31^{4} +O(31^{5})\)
$r_{ 7 }$	$=$	\( 6 a + 9 + \left(24 a + 9\right)\cdot 31 + \left(8 a + 26\right)\cdot 31^{2} + \left(24 a + 4\right)\cdot 31^{3} + \left(26 a + 26\right)\cdot 31^{4} +O(31^{5})\)

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 values
			$c1$
$1$	$1$	$()$	$21$
$21$	$2$	$(1,2)$	$1$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-3$
$105$	$2$	$(1,2)(3,4)$	$1$
$70$	$3$	$(1,2,3)$	$-3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$-1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.