Artin representation 1.751.2t1.a.a

• Label: 1.751.2t1.a.a
• Dimension: $1$
• Group: $C_2$
• Conductor: $751$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$1$
Group:	$C_2$
Conductor:	\(751\)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin field:	\(\Q(\sqrt{-751}) \)
Galois orbit size:	$1$
Smallest permutation container:	$C_2$
Parity:	odd
Dirichlet character:	\(\displaystyle\left(\frac{-751}{\bullet}\right)\)
Projective image:	$C_1$
Projective field:	\(\Q\)

Defining polynomial

$f(x)$

$=$

\(x^{2} - x + 188\)  .

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

Roots:

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

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
$1$	$1$	$()$	$1$
$1$	$2$	$(1,2)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.