Properties

Label 2.1399.9t3.a.a
Dimension $2$
Group $D_{9}$
Conductor $1399$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{9}$
Conductor: \(1399\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 9.1.3830635754401.1
Galois orbit size: $3$
Smallest permutation container: $D_{9}$
Parity: odd
Determinant: 1.1399.2t1.a.a
Projective image: $D_9$
Projective stem field: 9.1.3830635754401.1

Defining polynomial

$f(x)$$=$\(x^{9} - 3 x^{8} + 3 x^{7} + 13 x^{6} - 21 x^{5} - 4 x^{4} + 65 x^{3} - 56 x^{2} + 10 x - 9\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \(x^{3} + 2 x + 18\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 22 a^{2} + 20 a + 22 + \left(6 a^{2} + 17 a + 16\right)\cdot 23 + \left(4 a^{2} + 6 a + 5\right)\cdot 23^{2} + \left(4 a^{2} + 11 a + 13\right)\cdot 23^{3} + \left(22 a^{2} + 13 a + 6\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 4 a^{2} + 11 a + 21 + \left(8 a^{2} + 14 a + 10\right)\cdot 23 + \left(6 a^{2} + 13 a + 8\right)\cdot 23^{2} + \left(6 a^{2} + 19 a + 8\right)\cdot 23^{3} + \left(6 a^{2} + 17 a + 8\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 a^{2} + 4 a + 18 + \left(12 a^{2} + 15 a + 1\right)\cdot 23 + \left(12 a^{2} + 12 a + 9\right)\cdot 23^{2} + \left(10 a^{2} + 3 a + 6\right)\cdot 23^{3} + \left(21 a^{2} + 13 a + 13\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 3 a^{2} + 18 a + 12 + 6 a^{2}23 + \left(10 a^{2} + 17 a + 6\right)\cdot 23^{2} + \left(19 a^{2} + 8 a + 18\right)\cdot 23^{3} + \left(2 a^{2} + 13 a + 3\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 6 a^{2} + 3 a + 16 + \left(20 a^{2} + 11\right)\cdot 23 + \left(8 a^{2} + 13 a + 19\right)\cdot 23^{2} + \left(18 a^{2} + a + 16\right)\cdot 23^{3} + \left(18 a^{2} + 19 a + 9\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + 8 + \left(2 a^{2} + 16 a + 10\right)\cdot 23 + \left(4 a^{2} + 19 a + 5\right)\cdot 23^{2} + \left(6 a^{2} + 22 a + 8\right)\cdot 23^{3} + \left(18 a^{2} + 14 a + 1\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 21 a^{2} + 8 a + 13 + \left(9 a^{2} + 4 a + 5\right)\cdot 23 + \left(8 a^{2} + 22 a + 11\right)\cdot 23^{2} + \left(22 a^{2} + 2 a + 14\right)\cdot 23^{3} + \left(20 a^{2} + 19 a + 12\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 16 a^{2} + 9 a + 14 + \left(12 a^{2} + 21 a + 1\right)\cdot 23 + \left(2 a^{2} + 14 a + 11\right)\cdot 23^{2} + \left(22 a^{2} + a + 6\right)\cdot 23^{3} + \left(7 a^{2} + 4 a + 18\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( a^{2} + 11 a + 17 + \left(13 a^{2} + a + 9\right)\cdot 23 + \left(11 a^{2} + 18 a + 15\right)\cdot 23^{2} + \left(5 a^{2} + 19 a + 22\right)\cdot 23^{3} + \left(19 a^{2} + 22 a + 17\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$2$
$9$$2$$(1,2)(3,4)(5,8)(6,7)$$0$
$2$$3$$(1,4,7)(2,6,3)(5,9,8)$$-1$
$2$$9$$(1,9,2,4,8,6,7,5,3)$$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$
$2$$9$$(1,2,8,7,3,9,4,6,5)$$\zeta_{9}^{5} + \zeta_{9}^{4}$
$2$$9$$(1,8,3,4,5,2,7,9,6)$$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$

The blue line marks the conjugacy class containing complex conjugation.