Artin representation 15.1399e5_21911e5.42t412.1c1

• Label: 15.1399e5_21911e5.42t412.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 1399^{5} \cdot 21911^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$27064472490190296689173241511691806449= 1399^{5} \cdot 21911^{5} $
Artin number field:	Splitting field of $f= x^{7} - 7 x^{5} - x^{4} + 11 x^{3} + 3 x^{2} - 3 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T412
Parity:	Even
Determinant:	1.1399_21911.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 80 a + \left(32 a + 59\right)\cdot 83 + \left(13 a + 79\right)\cdot 83^{2} + \left(48 a + 56\right)\cdot 83^{3} + \left(75 a + 29\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 3 a + 80 + \left(50 a + 11\right)\cdot 83 + \left(69 a + 60\right)\cdot 83^{2} + \left(34 a + 8\right)\cdot 83^{3} + \left(7 a + 57\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 26 a + 80 + \left(64 a + 75\right)\cdot 83 + \left(48 a + 7\right)\cdot 83^{2} + \left(27 a + 4\right)\cdot 83^{3} + \left(27 a + 4\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 18 + 72\cdot 83 + 79\cdot 83^{2} + 83^{3} + 52\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 6 + 44\cdot 83 + 66\cdot 83^{2} + 59\cdot 83^{3} + 15\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 57 a + 23 + \left(18 a + 31\right)\cdot 83 + \left(34 a + 75\right)\cdot 83^{2} + \left(55 a + 65\right)\cdot 83^{3} + \left(55 a + 3\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 42 + 37\cdot 83 + 45\cdot 83^{2} + 51\cdot 83^{3} + 3\cdot 83^{4} +O\left(83^{ 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.