Artin representation 1.7.3t1.a.b

• Label: 1.7.3t1.a.b
• Dimension: $1$
• Group: $C_3$
• Conductor: $7$
• Root number: not computed
• Indicator: $0$

Basic invariants

Dimension:	$1$
Group:	$C_3$
Conductor:	\(7\)
Artin field:	\(\Q(\zeta_{7})^+\)
Galois orbit size:	$2$
Smallest permutation container:	$C_3$
Parity:	even
Dirichlet character:	\(\chi_{7}(4,\cdot)\)
Projective image:	$C_1$
Projective field:	\(\Q\)

Defining polynomial

$f(x)$

$=$

\(x^{3} - x^{2} - 2 x + 1\)  .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	\( 3 + 2\cdot 13 + 5\cdot 13^{2} + 5\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$	$=$	\( 5 + 10\cdot 13 + 3\cdot 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\)
$r_{ 3 }$	$=$	\( 6 + 4\cdot 13^{2} + 12\cdot 13^{3} + 13^{4} +O(13^{5})\)

Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $

Cycle notation

$(1,2,3)$

Character values on conjugacy classes

Size	Order	Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $	Character value
$1$	$1$	$()$	$1$
$1$	$3$	$(1,2,3)$	$-\zeta_{3} - 1$
$1$	$3$	$(1,3,2)$	$\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.