Artin representation 15.313e10_176713e10.42t411.1c1

• Label: 15.313e10_176713e10.42t411.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 313^{10} \cdot 176713^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$267996038077100889941118687428784970426950215576903413757053200765185625628801= 313^{10} \cdot 176713^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 7 x^{5} + 8 x^{4} + 10 x^{3} - 11 x^{2} - 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 $\Q_{ 311 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 311 }$: $ x^{2} + 310 x + 17 $

Roots:

$r_{ 1 }$	$=$	$ 49 a + 59 + \left(122 a + 217\right)\cdot 311 + \left(123 a + 254\right)\cdot 311^{2} + \left(310 a + 253\right)\cdot 311^{3} + \left(29 a + 44\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 262 a + 108 + \left(188 a + 290\right)\cdot 311 + \left(187 a + 255\right)\cdot 311^{2} + 129\cdot 311^{3} + \left(281 a + 75\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 218 a + 271 + \left(261 a + 66\right)\cdot 311 + \left(163 a + 90\right)\cdot 311^{2} + \left(74 a + 82\right)\cdot 311^{3} + \left(17 a + 273\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 64 + 34\cdot 311 + 94\cdot 311^{2} + 169\cdot 311^{3} + 85\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 93 a + 178 + \left(49 a + 110\right)\cdot 311 + \left(147 a + 303\right)\cdot 311^{2} + \left(236 a + 303\right)\cdot 311^{3} + \left(293 a + 215\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 261 + 263\cdot 311 + 79\cdot 311^{2} + 169\cdot 311^{3} + 9\cdot 311^{4} +O\left(311^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 304 + 260\cdot 311 + 165\cdot 311^{2} + 135\cdot 311^{3} + 228\cdot 311^{4} +O\left(311^{ 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.