Properties

Label 1.3e2.3t1.1c2
Dimension 1
Group $C_3$
Conductor $ 3^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:$9= 3^{2} $
Artin number field: Splitting field of $f= x^{3} - 3 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_3$
Parity: Even
Corresponding Dirichlet character: \(\chi_{9}(4,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 3 + 12\cdot 17 + 14\cdot 17^{2} + 14\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 2 } &= 4 + 9\cdot 17 + 13\cdot 17^{2} + 10\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 3 } &= 10 + 12\cdot 17 + 5\cdot 17^{2} + 8\cdot 17^{3} + 13\cdot 17^{4} +O\left(17^{ 5 }\right) \\ \end{aligned}\]

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$
$1$$3$$(1,3,2)$$\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.