Properties

Label 1.19_11149.2t1.1c1
Dimension 1
Group $C_2$
Conductor $ 19 \cdot 11149 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$1$
Group:$C_2$
Conductor:$211831= 19 \cdot 11149 $
Artin number field: Splitting field of $f=x^{2} - x + 52958$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2$
Parity: Odd
Corresponding Dirichlet character: \(\displaystyle\left(\frac{-211831}{\bullet}\right)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 257 +O\left(5^{ 5 }\right) \\ r_{ 2 } &= -256 +O\left(5^{ 5 }\right) \\ \end{aligned}\]

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

Cycle notation
$(1,2)$

Character values on conjugacy classes

SizeOrderAction 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.