Artin representation 15.23e10_14369e10.42t411.1c1

• Label: 15.23e10_14369e10.42t411.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 23^{10} \cdot 14369^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$15543320708312223666487426847706973574008825175391976849= 23^{10} \cdot 14369^{10} $
Artin number field:	Splitting field of $f= x^{7} + x^{5} - x^{3} - x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T411
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 521 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 521 }$: $ x^{2} + 515 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 488 + 5\cdot 521 + 3\cdot 521^{2} + 208\cdot 521^{3} + 195\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 169 a + 518 + \left(14 a + 437\right)\cdot 521 + \left(326 a + 282\right)\cdot 521^{2} + \left(308 a + 342\right)\cdot 521^{3} + \left(289 a + 361\right)\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 155 + 361\cdot 521 + 468\cdot 521^{2} + 423\cdot 521^{3} + 373\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 352 a + 490 + \left(506 a + 354\right)\cdot 521 + \left(194 a + 140\right)\cdot 521^{2} + \left(212 a + 305\right)\cdot 521^{3} + \left(231 a + 227\right)\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 199 + 373\cdot 521 + 481\cdot 521^{2} + 3\cdot 521^{3} + 235\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 327 + 10\cdot 521 + 31\cdot 521^{2} + 419\cdot 521^{3} + 143\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 428 + 18\cdot 521 + 155\cdot 521^{2} + 381\cdot 521^{3} + 25\cdot 521^{4} +O\left(521^{ 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.