Artin representation 15.919969e10.42t411.1c1

• Label: 15.919969e10.42t411.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 919969^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$434242106393169335125215445637539215905456226188558113780801= 919969^{10} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} - x^{5} + 3 x^{4} + x^{3} - 3 x^{2} - 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 $\Q_{ 67 }$ to precision 5.

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

Roots:

$r_{ 1 }$	$=$	$ 40 a + 14 + \left(32 a + 37\right)\cdot 67 + \left(36 a + 18\right)\cdot 67^{2} + \left(13 a + 53\right)\cdot 67^{3} + \left(57 a + 62\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 28 a + 17 + \left(32 a + 65\right)\cdot 67 + \left(38 a + 55\right)\cdot 67^{2} + \left(6 a + 3\right)\cdot 67^{3} + \left(8 a + 15\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 12 + 25\cdot 67 + 15\cdot 67^{2} + 15\cdot 67^{3} + 54\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 39 a + 62 + \left(34 a + 32\right)\cdot 67 + \left(28 a + 43\right)\cdot 67^{2} + \left(60 a + 58\right)\cdot 67^{3} + \left(58 a + 40\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 47 + 63\cdot 67 + 30\cdot 67^{2} + 62\cdot 67^{3} + 30\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 27 a + 40 + \left(34 a + 60\right)\cdot 67 + \left(30 a + 64\right)\cdot 67^{2} + \left(53 a + 3\right)\cdot 67^{3} + \left(9 a + 10\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 11 + 50\cdot 67 + 38\cdot 67^{2} + 3\cdot 67^{3} + 54\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.