Artin representation 1.112531.2t1.a.a

• Label: 1.112531.2t1.a.a
• Dimension: $1$
• Group: $C_2$
• Conductor: $112531$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$1$
Group:	$C_2$
Conductor:	$112531$$\medspace = 43 \cdot 2617$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin field:	Galois closure of $\Q(\sqrt{-112531})$
Galois orbit size:	$1$
Smallest permutation container:	$C_2$
Parity:	odd
Dirichlet character:	$\displaystyle\left(\frac{-112531}{\bullet}\right)$
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Defining polynomial

$f(x)$

$=$

$x^{2} - x + 28133$ .

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

Roots:

$r_{ 1 }$	$=$	$2 + 5 + 5^{2} + 4\cdot 5^{3} + 5^{4} +O(5^{5})$
$r_{ 2 }$	$=$	$4 + 3\cdot 5 + 3\cdot 5^{2} + 3\cdot 5^{4} +O(5^{5})$

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

Cycle notation

$(1,2)$

Character values on conjugacy classes

Size	Order	Action on $r_{ 1 }, r_{ 2 }$	Character value	Complex conjugation
$1$	$1$	$()$	$1$
$1$	$2$	$(1,2)$	$-1$	✓