Artin representation 15.11e10_577e10_11003e10.42t411.1c1

• Label: 15.11e10_577e10_11003e10.42t411.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 11^{10} \cdot 577^{10} \cdot 11003^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$2759282158024921618666632360936482451102955205765424285479931351638874634112401= 11^{10} \cdot 577^{10} \cdot 11003^{10} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{6} - 4 x^{5} + 16 x^{4} - 20 x^{2} + 8 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 $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 4 a + 28 + \left(10 a + 30\right)\cdot 31 + \left(17 a + 23\right)\cdot 31^{2} + \left(13 a + 10\right)\cdot 31^{3} + \left(21 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 29 + 17\cdot 31^{2} + 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 5 a + 2 + \left(16 a + 4\right)\cdot 31 + 26\cdot 31^{2} + \left(14 a + 24\right)\cdot 31^{3} + \left(13 a + 24\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 26 a + 12 + 14 a\cdot 31 + \left(30 a + 11\right)\cdot 31^{2} + \left(16 a + 21\right)\cdot 31^{3} + \left(17 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 14 + 16\cdot 31 + 10\cdot 31^{2} + 22\cdot 31^{3} + 5\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 27 a + 5 + \left(20 a + 16\right)\cdot 31 + \left(13 a + 17\right)\cdot 31^{2} + \left(17 a + 20\right)\cdot 31^{3} + \left(9 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 6 + 24\cdot 31 + 17\cdot 31^{2} + 22\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 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.