Properties

Label 1.3e3.9t1.1c4
Dimension 1
Group $C_9$
Conductor $ 3^{3}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_9$
Conductor:$27= 3^{3} $
Artin number field: Splitting field of $f= x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1 $ over $\Q$
Size of Galois orbit: 6
Smallest containing permutation representation: $C_9$
Parity: Even
Corresponding Dirichlet character: \(\chi_{27}(4,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{3} + x + 14 $
Roots:
$r_{ 1 }$ $=$ $ 16 a^{2} + 15 a + 5 + \left(12 a^{2} + 14 a + 14\right)\cdot 17 + \left(2 a^{2} + 11 a + 1\right)\cdot 17^{2} + \left(5 a^{2} + 5 a + 9\right)\cdot 17^{3} + \left(16 a^{2} + 8 a + 16\right)\cdot 17^{4} + \left(a^{2} + 14 a + 6\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 13 a^{2} + 2 a + 3 + \left(6 a^{2} + 7 a + 10\right)\cdot 17 + \left(14 a^{2} + 13 a + 9\right)\cdot 17^{2} + \left(2 a^{2} + 3 a + 7\right)\cdot 17^{3} + \left(10 a + 11\right)\cdot 17^{4} + \left(13 a^{2} + 7 a + 8\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 13 a^{2} + 13 a + 3 + \left(14 a^{2} + 7 a + 4\right)\cdot 17 + \left(13 a^{2} + 4 a + 9\right)\cdot 17^{2} + \left(12 a^{2} + 8\right)\cdot 17^{3} + \left(8 a^{2} + 11 a + 11\right)\cdot 17^{4} + \left(8 a^{2} + 5 a + 5\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 14 a^{2} + 15 + \left(a^{2} + 3 a + 6\right)\cdot 17 + \left(14 a^{2} + 8 a + 9\right)\cdot 17^{2} + \left(a^{2} + 3 a + 12\right)\cdot 17^{3} + \left(12 a^{2} + 13\right)\cdot 17^{4} + \left(16 a^{2} + 2 a + 16\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 7 a^{2} + 8 a + 16 + \left(6 a^{2} + 6 a + 9\right)\cdot 17 + \left(7 a^{2} + 6 a + 10\right)\cdot 17^{2} + \left(15 a^{2} + 16 a + 4\right)\cdot 17^{3} + \left(14 a^{2} + 15 a + 4\right)\cdot 17^{4} + \left(a^{2} + 15 a + 1\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 2 a^{2} + 9 a + 7 + \left(7 a^{2} + a + 10\right)\cdot 17 + \left(4 a^{2} + a + 8\right)\cdot 17^{2} + \left(13 a^{2} + 3 a + 14\right)\cdot 17^{3} + \left(4 a^{2} + 6 a + 8\right)\cdot 17^{4} + \left(16 a^{2} + 4 a + 16\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 14 a^{2} + 7 a + 15 + \left(3 a^{2} + 3 a + 13\right)\cdot 17 + \left(12 a^{2} + 14 a + 13\right)\cdot 17^{2} + \left(15 a^{2} + 13 a + 4\right)\cdot 17^{3} + \left(a^{2} + 7 a + 1\right)\cdot 17^{4} + \left(2 a^{2} + 10 a + 7\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 16 a^{2} + 10 a + 5 + \left(13 a^{2} + 9\right)\cdot 17 + \left(9 a^{2} + 4 a + 6\right)\cdot 17^{2} + \left(15 a^{2} + 8 a + 10\right)\cdot 17^{3} + \left(12 a^{2} + 2 a + 8\right)\cdot 17^{4} + \left(15 a^{2} + 15 a + 10\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 9 }$ $=$ $ 7 a^{2} + 4 a + 16 + \left(6 a + 5\right)\cdot 17 + \left(6 a^{2} + 4 a + 15\right)\cdot 17^{2} + \left(2 a^{2} + 13 a + 12\right)\cdot 17^{3} + \left(13 a^{2} + 5 a + 8\right)\cdot 17^{4} + \left(8 a^{2} + 9 a + 11\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(1,7,9,8,2,3,6,5,4)$
$(1,8,6)(2,5,7)(3,4,9)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$1$
$1$$3$$(1,8,6)(2,5,7)(3,4,9)$$-\zeta_{9}^{3} - 1$
$1$$3$$(1,6,8)(2,7,5)(3,9,4)$$\zeta_{9}^{3}$
$1$$9$$(1,7,9,8,2,3,6,5,4)$$\zeta_{9}^{5}$
$1$$9$$(1,9,2,6,4,7,8,3,5)$$\zeta_{9}$
$1$$9$$(1,2,4,8,5,9,6,7,3)$$\zeta_{9}^{2}$
$1$$9$$(1,3,7,6,9,5,8,4,2)$$-\zeta_{9}^{4} - \zeta_{9}$
$1$$9$$(1,5,3,8,7,4,6,2,9)$$-\zeta_{9}^{5} - \zeta_{9}^{2}$
$1$$9$$(1,4,5,6,3,2,8,9,7)$$\zeta_{9}^{4}$
The blue line marks the conjugacy class containing complex conjugation.