Properties

Label 1.227287.2t1.a.a
Dimension 1
Group $C_2$
Conductor $ 167 \cdot 1361 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$1$
Group:$C_2$
Conductor:$227287= 167 \cdot 1361 $
Artin number field: Splitting field of \(\Q(\sqrt{-227287}) \) defined by $f= x^{2} - x + 56822 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2$
Parity: Odd
Corresponding Dirichlet character: \(\displaystyle\left(\frac{-227287}{\bullet}\right)\)
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 13 + 17\cdot 31 + 19\cdot 31^{2} + 14\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 + 13\cdot 31 + 11\cdot 31^{2} + 16\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$

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.