Galois orbit of Artin representations 2.5e2_7e2.6t5.1

• Label: 2.5e2_7e2.6t5.1
• Dimension: 2
• Group: $S_3\times C_3$
• Conductor: $ 5^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$S_3\times C_3$
Conductor:	$1225= 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{6} - x^{5} + x^{4} + 6 x^{3} + 8 x^{2} - 29 x + 22 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$S_3\times C_3$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $

Roots:

$r_{ 1 }$	$=$	$ 31 a + 31 + \left(25 a + 6\right)\cdot 41 + \left(15 a + 25\right)\cdot 41^{2} + \left(28 a + 38\right)\cdot 41^{3} + \left(7 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 5 a + 33 + \left(11 a + 10\right)\cdot 41 + \left(27 a + 29\right)\cdot 41^{2} + \left(23 a + 9\right)\cdot 41^{3} + \left(35 a + 23\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 30 a + 22 + \left(17 a + 15\right)\cdot 41 + \left(8 a + 39\right)\cdot 41^{2} + \left(34 a + 11\right)\cdot 41^{3} + \left(23 a + 28\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 11 a + 30 + \left(23 a + 38\right)\cdot 41 + \left(32 a + 5\right)\cdot 41^{2} + \left(6 a + 24\right)\cdot 41^{3} + \left(17 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 10 a + 1 + \left(15 a + 12\right)\cdot 41 + \left(25 a + 5\right)\cdot 41^{2} + \left(12 a + 26\right)\cdot 41^{3} + \left(33 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 36 a + 7 + \left(29 a + 39\right)\cdot 41 + \left(13 a + 17\right)\cdot 41^{2} + \left(17 a + 12\right)\cdot 41^{3} + \left(5 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation

$(1,5)(2,6)(3,4)$

$(3,6,5)$

$(1,4,2)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$3$	$2$	$(1,5)(2,6)(3,4)$	$0$	$0$
$1$	$3$	$(1,4,2)(3,6,5)$	$2 \zeta_{3}$	$-2 \zeta_{3} - 2$
$1$	$3$	$(1,2,4)(3,5,6)$	$-2 \zeta_{3} - 2$	$2 \zeta_{3}$
$2$	$3$	$(3,6,5)$	$\zeta_{3} + 1$	$-\zeta_{3}$
$2$	$3$	$(3,5,6)$	$-\zeta_{3}$	$\zeta_{3} + 1$
$2$	$3$	$(1,4,2)(3,5,6)$	$-1$	$-1$
$3$	$6$	$(1,5,4,3,2,6)$	$0$	$0$
$3$	$6$	$(1,6,2,3,4,5)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.