Artin representation 15.79e10_9199e10.42t411.1c1

• Label: 15.79e10_9199e10.42t411.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 79^{10} \cdot 9199^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$41084414436021074786189022448905327434527286887533293155201= 79^{10} \cdot 9199^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} + 2 x^{4} - 5 x^{3} + 4 x^{2} - 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 $\Q_{ 433 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 433 }$: $ x^{2} + 432 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 193 + 119\cdot 433 + 23\cdot 433^{2} + 54\cdot 433^{3} + 159\cdot 433^{4} +O\left(433^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 34 + 204\cdot 433 + 9\cdot 433^{2} + 46\cdot 433^{3} + 400\cdot 433^{4} +O\left(433^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 294 a + 23 + \left(290 a + 234\right)\cdot 433 + \left(68 a + 158\right)\cdot 433^{2} + \left(405 a + 189\right)\cdot 433^{3} + \left(342 a + 204\right)\cdot 433^{4} +O\left(433^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 148 + 171\cdot 433 + 107\cdot 433^{2} + 75\cdot 433^{3} + 245\cdot 433^{4} +O\left(433^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 139 a + 317 + \left(142 a + 230\right)\cdot 433 + \left(364 a + 369\right)\cdot 433^{2} + \left(27 a + 92\right)\cdot 433^{3} + \left(90 a + 142\right)\cdot 433^{4} +O\left(433^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 206 a + 406 + \left(196 a + 390\right)\cdot 433 + \left(426 a + 416\right)\cdot 433^{2} + \left(223 a + 88\right)\cdot 433^{3} + \left(392 a + 206\right)\cdot 433^{4} +O\left(433^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 227 a + 179 + \left(236 a + 381\right)\cdot 433 + \left(6 a + 213\right)\cdot 433^{2} + \left(209 a + 319\right)\cdot 433^{3} + \left(40 a + 374\right)\cdot 433^{4} +O\left(433^{ 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.