Properties

Label 1.163.3t1.1c1
Dimension 1
Group $C_3$
Conductor $ 163 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:$163 $
Artin number field: Splitting field of $f=x^{3} - x^{2} - 54 x + 169$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_3$
Parity: Even
Corresponding Dirichlet character: \(\chi_{163}(58,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 556 +O\left(5^{ 5 }\right) \\ r_{ 2 } &= 1392 +O\left(5^{ 5 }\right) \\ r_{ 3 } &= 1178 +O\left(5^{ 5 }\right) \\ \end{aligned}\]

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

Cycle notation
$(1,2,3)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ Character value
$1$$1$$()$$1$
$1$$3$$(1,2,3)$$\zeta_{3}$
$1$$3$$(1,3,2)$$-\zeta_{3} - 1$
The blue line marks the conjugacy class containing complex conjugation.