Properties

Label 1.189.9t1.b.a
Dimension $1$
Group $C_9$
Conductor $189$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_9$
Conductor: \(189\)\(\medspace = 3^{3} \cdot 7 \)
Artin number field: Galois closure of 9.9.3691950281939241.2
Galois orbit size: $6$
Smallest permutation container: $C_9$
Parity: even
Dirichlet character: \(\chi_{189}(121,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831 $.

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{3} + 6 x + 35 $

Roots:
$r_{ 1 }$ $=$ $ 13 a^{2} + 29 a + 15 + \left(6 a^{2} + 9 a + 25\right)\cdot 37 + \left(28 a^{2} + 1\right)\cdot 37^{2} + \left(25 a^{2} + 5 a + 29\right)\cdot 37^{3} + \left(34 a^{2} + 26 a + 27\right)\cdot 37^{4} + \left(14 a^{2} + 24 a + 22\right)\cdot 37^{5} + \left(8 a^{2} + 8 a + 33\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 21 a^{2} + 35 a + 10 + \left(29 a^{2} + 5 a + 7\right)\cdot 37 + \left(9 a^{2} + 8 a + 2\right)\cdot 37^{2} + \left(24 a^{2} + 33 a + 23\right)\cdot 37^{3} + \left(34 a^{2} + 14 a + 27\right)\cdot 37^{4} + \left(22 a + 3\right)\cdot 37^{5} + \left(22 a^{2} + 20 a + 14\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 19 a^{2} + 26 a + 2 + \left(a^{2} + 28 a + 6\right)\cdot 37 + \left(32 a^{2} + 9 a + 17\right)\cdot 37^{2} + \left(3 a^{2} + 9 a + 15\right)\cdot 37^{3} + \left(30 a^{2} + 34 a + 9\right)\cdot 37^{4} + \left(27 a^{2} + 2 a\right)\cdot 37^{5} + \left(34 a^{2} + 36 a + 28\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 8 a^{2} + 27 a + 32 + \left(11 a^{2} + 24 a + 7\right)\cdot 37 + \left(16 a^{2} + 23 a + 28\right)\cdot 37^{2} + \left(36 a^{2} + 23 a + 34\right)\cdot 37^{3} + \left(11 a^{2} + 25 a + 10\right)\cdot 37^{4} + \left(9 a^{2} + 7 a\right)\cdot 37^{5} + \left(31 a^{2} + 28 a + 14\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 20 a^{2} + 2 a + 6 + \left(31 a^{2} + 4 a + 15\right)\cdot 37 + \left(28 a^{2} + 6 a + 4\right)\cdot 37^{2} + \left(5 a^{2} + 20 a + 23\right)\cdot 37^{3} + \left(29 a^{2} + 9 a + 5\right)\cdot 37^{4} + \left(18 a^{2} + 22 a + 1\right)\cdot 37^{5} + \left(7 a^{2} + 21 a + 30\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 23 a^{2} + 18 + \left(a^{2} + 2 a + 6\right)\cdot 37 + \left(5 a^{2} + 17 a + 20\right)\cdot 37^{2} + \left(6 a^{2} + 7 a + 24\right)\cdot 37^{3} + \left(17 a^{2} + 30 a + 31\right)\cdot 37^{4} + \left(6 a^{2} + 4 a + 25\right)\cdot 37^{5} + \left(14 a^{2} + 33 a + 19\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 3 a^{2} + 10 a + 12 + \left(a^{2} + 21 a + 4\right)\cdot 37 + \left(36 a^{2} + 28 a + 33\right)\cdot 37^{2} + \left(23 a^{2} + 35 a + 21\right)\cdot 37^{3} + \left(4 a^{2} + 32 a + 18\right)\cdot 37^{4} + \left(21 a^{2} + 26 a + 10\right)\cdot 37^{5} + \left(6 a^{2} + 7 a + 26\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 9 a^{2} + 8 a + 36 + \left(31 a^{2} + 8 a + 13\right)\cdot 37 + \left(28 a^{2} + 7 a + 4\right)\cdot 37^{2} + \left(31 a^{2} + 30 a + 16\right)\cdot 37^{3} + \left(32 a^{2} + a + 20\right)\cdot 37^{4} + \left(8 a^{2} + 7 a + 35\right)\cdot 37^{5} + \left(35 a^{2} + 24 a + 29\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 9 }$ $=$ $ 32 a^{2} + 11 a + 17 + \left(33 a^{2} + 6 a + 24\right)\cdot 37 + \left(36 a^{2} + 10 a + 36\right)\cdot 37^{2} + \left(26 a^{2} + 20 a + 33\right)\cdot 37^{3} + \left(26 a^{2} + 9 a + 32\right)\cdot 37^{4} + \left(2 a^{2} + 29 a + 10\right)\cdot 37^{5} + \left(25 a^{2} + 4 a + 26\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$1$
$1$$3$$(1,7,2)(3,9,6)(4,8,5)$$\zeta_{9}^{3}$
$1$$3$$(1,2,7)(3,6,9)(4,5,8)$$-\zeta_{9}^{3} - 1$
$1$$9$$(1,4,9,7,8,6,2,5,3)$$\zeta_{9}$
$1$$9$$(1,9,8,2,3,4,7,6,5)$$\zeta_{9}^{2}$
$1$$9$$(1,8,3,7,5,9,2,4,6)$$\zeta_{9}^{4}$
$1$$9$$(1,6,4,2,9,5,7,3,8)$$\zeta_{9}^{5}$
$1$$9$$(1,5,6,7,4,3,2,8,9)$$-\zeta_{9}^{4} - \zeta_{9}$
$1$$9$$(1,3,5,2,6,8,7,9,4)$$-\zeta_{9}^{5} - \zeta_{9}^{2}$

The blue line marks the conjugacy class containing complex conjugation.