Properties

Label 1.9.3t1.a.a
Dimension $1$
Group $C_3$
Conductor $9$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_3$
Conductor: \(9\)\(\medspace = 3^{2}\)
Artin number field: Galois closure of \(\Q(\zeta_{9})^+\)
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Dirichlet character: \(\chi_{9}(7,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{3} - 3 x - 1 $.

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 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)$

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.