Artin representation 15.41e5_1713281e5.42t412.1c1

• Label: 15.41e5_1713281e5.42t412.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 41^{5} \cdot 1713281^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$1710260545165616552306502663328809606601= 41^{5} \cdot 1713281^{5} $
Artin number field:	Splitting field of $f= x^{7} - 7 x^{5} - x^{4} + 13 x^{3} + 2 x^{2} - 6 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T412
Parity:	Even
Determinant:	1.41_1713281.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $ x^{2} + 274 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 219 a + 56 + \left(197 a + 56\right)\cdot 277 + \left(158 a + 117\right)\cdot 277^{2} + \left(201 a + 38\right)\cdot 277^{3} + \left(203 a + 196\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 152 + 202\cdot 277 + 96\cdot 277^{2} + 256\cdot 277^{3} + 43\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 222 + 213\cdot 277 + 231\cdot 277^{2} + 36\cdot 277^{3} + 253\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 273 + 88\cdot 277 + 172\cdot 277^{2} + 89\cdot 277^{3} + 191\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 271 a + 132 + \left(276 a + 193\right)\cdot 277 + \left(48 a + 250\right)\cdot 277^{2} + \left(103 a + 247\right)\cdot 277^{3} + \left(250 a + 138\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 58 a + 159 + \left(79 a + 153\right)\cdot 277 + \left(118 a + 118\right)\cdot 277^{2} + \left(75 a + 207\right)\cdot 277^{3} + \left(73 a + 51\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 6 a + 114 + 199\cdot 277 + \left(228 a + 120\right)\cdot 277^{2} + \left(173 a + 231\right)\cdot 277^{3} + \left(26 a + 232\right)\cdot 277^{4} +O\left(277^{ 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
$1$	$1$	$()$	$15$
$21$	$2$	$(1,2)$	$5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-3$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.