Artin representation 3.2e6_7e2_11e2.12t33.1c2

• Label: 3.2e6_7e2_11e2.12t33.1c2
• Dimension: 3
• Group: $A_5$
• Conductor: $ 2^{6} \cdot 7^{2} \cdot 11^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$A_5$
Conductor:	$379456= 2^{6} \cdot 7^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{5} - x^{4} + 4 x^{3} - 8 x^{2} + x + 7 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$A_5$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 503 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 125 + 151\cdot 503 + 114\cdot 503^{2} + 427\cdot 503^{3} + 483\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 162 + 31\cdot 503 + 202\cdot 503^{2} + 473\cdot 503^{3} + 305\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 190 + 102\cdot 503 + 88\cdot 503^{2} + 398\cdot 503^{3} + 441\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 244 + 130\cdot 503 + 186\cdot 503^{2} + 54\cdot 503^{3} + 435\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 286 + 87\cdot 503 + 415\cdot 503^{2} + 155\cdot 503^{3} + 345\cdot 503^{4} +O\left(503^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation

$(1,2,3)$

$(3,4,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 5 }$	Character value
$1$	$1$	$()$	$3$
$15$	$2$	$(1,2)(3,4)$	$-1$
$20$	$3$	$(1,2,3)$	$0$
$12$	$5$	$(1,2,3,4,5)$	$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$	$5$	$(1,3,4,5,2)$	$-\zeta_{5}^{3} - \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.