Properties

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

Related objects

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:$163 $
Artin number field: Splitting field of 3.3.26569.1 defined by $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: \(\chi_{163}(58,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 1 + 5 + 2\cdot 5^{2} + 4\cdot 5^{3} +O\left(5^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 + 3\cdot 5 + 5^{3} + 2\cdot 5^{4} +O\left(5^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 + 2\cdot 5^{2} + 4\cdot 5^{3} + 5^{4} +O\left(5^{ 5 }\right)$

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.