Artin representation 21.5e11_53e11_89e11_263e11.42t418.1c1

• Label: 21.5e11_53e11_89e11_263e11.42t418.1c1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 5^{11} \cdot 53^{11} \cdot 89^{11} \cdot 263^{11}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$523007503650505502663037769689445129774176781783048149453009329139208984375= 5^{11} \cdot 53^{11} \cdot 89^{11} \cdot 263^{11} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + x^{5} + 4 x^{4} - 8 x^{3} + x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T418
Parity:	Odd
Determinant:	1.5_53_89_263.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 337 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 337 }$: $ x^{2} + 332 x + 10 $

Roots:

$r_{ 1 }$	$=$	$ 217 a + 224 + \left(256 a + 128\right)\cdot 337 + \left(161 a + 17\right)\cdot 337^{2} + \left(292 a + 202\right)\cdot 337^{3} + \left(157 a + 241\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 120 a + 298 + \left(80 a + 183\right)\cdot 337 + \left(175 a + 232\right)\cdot 337^{2} + \left(44 a + 154\right)\cdot 337^{3} + \left(179 a + 64\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 279 a + 85 + \left(108 a + 189\right)\cdot 337 + \left(306 a + 130\right)\cdot 337^{2} + \left(126 a + 190\right)\cdot 337^{3} + \left(131 a + 335\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 58 a + 132 + \left(228 a + 117\right)\cdot 337 + \left(30 a + 205\right)\cdot 337^{2} + \left(210 a + 181\right)\cdot 337^{3} + \left(205 a + 191\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 271 + 149\cdot 337 + 169\cdot 337^{2} + 84\cdot 337^{3} + 294\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 33 a + 256 + \left(265 a + 148\right)\cdot 337 + \left(137 a + 84\right)\cdot 337^{2} + \left(138 a + 327\right)\cdot 337^{3} + \left(50 a + 221\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 304 a + 84 + \left(71 a + 93\right)\cdot 337 + \left(199 a + 171\right)\cdot 337^{2} + \left(198 a + 207\right)\cdot 337^{3} + \left(286 a + 335\right)\cdot 337^{4} +O\left(337^{ 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
$1$	$1$	$()$	$21$
$21$	$2$	$(1,2)$	$-1$
$105$	$2$	$(1,2)(3,4)(5,6)$	$3$
$105$	$2$	$(1,2)(3,4)$	$1$
$70$	$3$	$(1,2,3)$	$-3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.