Galois orbit of Artin representations 21.37e10_199e10_461e10.84.1

• Label: 21.37e10_199e10_461e10.84.1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 37^{10} \cdot 199^{10} \cdot 461^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$203028618112401435703074107365887610708538184672291938047797283249= 37^{10} \cdot 199^{10} \cdot 461^{10} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} - 2 x^{5} + 6 x^{4} - x^{3} - 5 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	84
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 557 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 557 }$: $ x^{2} + 553 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 216 a + 461 + \left(406 a + 311\right)\cdot 557 + \left(321 a + 201\right)\cdot 557^{2} + \left(135 a + 296\right)\cdot 557^{3} + \left(52 a + 248\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 213 + 459\cdot 557 + 158\cdot 557^{2} + 262\cdot 557^{3} + 111\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 499 + 529\cdot 557 + 104\cdot 557^{2} + 58\cdot 557^{3} + 84\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 342 a + 428 + \left(496 a + 139\right)\cdot 557 + \left(521 a + 543\right)\cdot 557^{2} + \left(326 a + 508\right)\cdot 557^{3} + \left(312 a + 145\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 341 a + 211 + \left(150 a + 50\right)\cdot 557 + \left(235 a + 525\right)\cdot 557^{2} + \left(421 a + 516\right)\cdot 557^{3} + \left(504 a + 321\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 215 a + 125 + \left(60 a + 113\right)\cdot 557 + \left(35 a + 463\right)\cdot 557^{2} + \left(230 a + 180\right)\cdot 557^{3} + \left(244 a + 512\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 293 + 66\cdot 557 + 231\cdot 557^{2} + 404\cdot 557^{3} + 246\cdot 557^{4} +O\left(557^{ 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 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.