Artin representation 15.269e5_12619e5.42t412.1c1

• Label: 15.269e5_12619e5.42t412.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 269^{5} \cdot 12619^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$450698488695361902595330872033551= 269^{5} \cdot 12619^{5} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + x^{5} - x^{4} - 3 x^{3} + 4 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T412
Parity:	Odd
Determinant:	1.269_12619.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$: $ x^{2} + 220 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 55 a + 147 + \left(34 a + 23\right)\cdot 227 + \left(144 a + 79\right)\cdot 227^{2} + \left(72 a + 26\right)\cdot 227^{3} + \left(11 a + 11\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 172 a + 78 + \left(192 a + 208\right)\cdot 227 + \left(82 a + 145\right)\cdot 227^{2} + \left(154 a + 163\right)\cdot 227^{3} + \left(215 a + 17\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 24 + 75\cdot 227 + 6\cdot 227^{2} + 93\cdot 227^{3} + 121\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 69 + 19\cdot 227 + 162\cdot 227^{2} + 90\cdot 227^{3} + 180\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 32 a + 72 + \left(98 a + 80\right)\cdot 227 + \left(64 a + 162\right)\cdot 227^{2} + \left(65 a + 25\right)\cdot 227^{3} + \left(34 a + 70\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 224 + 219\cdot 227 + 63\cdot 227^{2} + 90\cdot 227^{3} + 35\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 195 a + 69 + \left(128 a + 54\right)\cdot 227 + \left(162 a + 61\right)\cdot 227^{2} + \left(161 a + 191\right)\cdot 227^{3} + \left(192 a + 17\right)\cdot 227^{4} +O\left(227^{ 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.