Artin representation 15.109e5_269e5_2153e5.42t412.1c1

• Label: 15.109e5_269e5_2153e5.42t412.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 109^{5} \cdot 269^{5} \cdot 2153^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$1002568462070463124405555645571930391793= 109^{5} \cdot 269^{5} \cdot 2153^{5} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 6 x^{3} - 7 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T412
Parity:	Even
Determinant:	1.109_269_2153.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 22 a + 51 + \left(46 a + 9\right)\cdot 67 + \left(35 a + 44\right)\cdot 67^{2} + 47 a\cdot 67^{3} + \left(42 a + 59\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 28 + 67 + 61\cdot 67^{2} + 8\cdot 67^{3} + 13\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 23 a + 18 + \left(64 a + 11\right)\cdot 67 + \left(39 a + 3\right)\cdot 67^{2} + \left(44 a + 28\right)\cdot 67^{3} + \left(12 a + 50\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 45 a + 5 + \left(20 a + 39\right)\cdot 67 + \left(31 a + 6\right)\cdot 67^{2} + \left(19 a + 21\right)\cdot 67^{3} + \left(24 a + 48\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 50 a + 63 + \left(7 a + 56\right)\cdot 67 + \left(2 a + 26\right)\cdot 67^{2} + \left(55 a + 46\right)\cdot 67^{3} + \left(41 a + 64\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 17 a + 62 + \left(59 a + 37\right)\cdot 67 + \left(64 a + 27\right)\cdot 67^{2} + \left(11 a + 63\right)\cdot 67^{3} + \left(25 a + 42\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 44 a + 43 + \left(2 a + 44\right)\cdot 67 + \left(27 a + 31\right)\cdot 67^{2} + \left(22 a + 32\right)\cdot 67^{3} + \left(54 a + 56\right)\cdot 67^{4} +O\left(67^{ 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.