Properties

Label 2.2003.9t3.1c3
Dimension 2
Group $D_{9}$
Conductor $ 2003 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{9}$
Conductor:$2003 $
Artin number field: Splitting field of $f= x^{9} - 3 x^{8} - 2 x^{7} + 24 x^{6} - 40 x^{5} - 88 x^{4} + 259 x^{3} + 75 x^{2} - 318 x + 148 $ over $\Q$
Size of Galois orbit: 3
Smallest containing permutation representation: $D_{9}$
Parity: Odd
Determinant: 1.2003.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{3} + 4 x + 17 $
Roots:
$r_{ 1 }$ $=$ $ 13 a^{2} + 12 a + 16 + \left(3 a^{2} + 9\right)\cdot 19 + \left(3 a^{2} + 4 a + 8\right)\cdot 19^{2} + \left(15 a^{2} + 11 a + 2\right)\cdot 19^{3} + \left(6 a^{2} + 4 a + 18\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 16 a^{2} + 14 a + 5 + \left(6 a^{2} + 10 a + 18\right)\cdot 19 + \left(11 a^{2} + 12 a + 4\right)\cdot 19^{2} + \left(5 a + 14\right)\cdot 19^{3} + \left(13 a^{2} + 15 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 10 a^{2} + 10 a + 8 + \left(3 a^{2} + 12 a + 9\right)\cdot 19 + \left(15 a^{2} + 2 a + 2\right)\cdot 19^{2} + \left(5 a^{2} + 18 a + 9\right)\cdot 19^{3} + \left(3 a^{2} + 9 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 a^{2} + 16 a + 1 + \left(3 a^{2} + 17 a + 15\right)\cdot 19 + \left(12 a^{2} + 13 a\right)\cdot 19^{2} + \left(11 a^{2} + 17 a + 12\right)\cdot 19^{3} + \left(6 a^{2} + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ a^{2} + 10 a + 3 + \left(12 a^{2} + 13\right)\cdot 19 + \left(3 a^{2} + a + 9\right)\cdot 19^{2} + \left(11 a^{2} + 9 a + 4\right)\cdot 19^{3} + \left(5 a^{2} + 13 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 17 a^{2} + 3 a + 14 + \left(14 a^{2} + 17 a + 1\right)\cdot 19 + \left(5 a^{2} + 15 a + 9\right)\cdot 19^{2} + \left(13 a^{2} + 16\right)\cdot 19^{3} + \left(2 a^{2} + 14 a\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 12 a^{2} + 10 a + 7 + \left(6 a^{2} + 2 a + 11\right)\cdot 19 + \left(4 a^{2} + 18 a + 11\right)\cdot 19^{2} + \left(6 a^{2} + 13 a + 16\right)\cdot 19^{3} +O\left(19^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 16 a^{2} + 18 a + 5 + \left(8 a^{2} + 3 a + 17\right)\cdot 19 + \left(18 a^{2} + 17 a + 4\right)\cdot 19^{2} + \left(6 a^{2} + 5 a + 12\right)\cdot 19^{3} + \left(15 a^{2} + 8 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 9 }$ $=$ $ 5 a^{2} + 2 a + 1 + \left(16 a^{2} + 10 a + 18\right)\cdot 19 + \left(a^{2} + 9 a + 4\right)\cdot 19^{2} + \left(5 a^{2} + 12 a + 7\right)\cdot 19^{3} + \left(3 a^{2} + 8 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$2$
$9$$2$$(1,5)(2,7)(3,6)(8,9)$$0$
$2$$3$$(1,5,4)(2,9,6)(3,8,7)$$-1$
$2$$9$$(1,8,9,5,7,6,4,3,2)$$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$
$2$$9$$(1,9,7,4,2,8,5,6,3)$$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$
$2$$9$$(1,7,2,5,3,9,4,8,6)$$\zeta_{9}^{5} + \zeta_{9}^{4}$
The blue line marks the conjugacy class containing complex conjugation.