Properties

Label 2.335.9t3.a.c
Dimension $2$
Group $D_{9}$
Conductor $335$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{9}$
Conductor: \(335\)\(\medspace = 5 \cdot 67 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 9.1.12594450625.1
Galois orbit size: $3$
Smallest permutation container: $D_{9}$
Parity: odd
Determinant: 1.335.2t1.a.a
Projective image: $D_9$
Projective stem field: 9.1.12594450625.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{3} + 2 x + 11\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a^{2} + 6 a + 1 + \left(a^{2} + a + 7\right)\cdot 13 + \left(4 a^{2} + 12 a + 7\right)\cdot 13^{2} + \left(7 a^{2} + 3 a + 12\right)\cdot 13^{3} + \left(5 a^{2} + 4 a + 6\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 11 a^{2} + 12 a + 6 + \left(11 a^{2} + a + 12\right)\cdot 13 + \left(6 a^{2} + 5 a + 6\right)\cdot 13^{2} + \left(11 a^{2} + 8 a + 9\right)\cdot 13^{3} + \left(11 a^{2} + 6 a + 6\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 a^{2} + 2 a + \left(12 a^{2} + 10 a + 5\right)\cdot 13 + \left(a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(8 a^{2} + 8 a + 7\right)\cdot 13^{3} + \left(10 a^{2} + 6 a + 4\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 6 a^{2} + a + 12 + \left(a^{2} + 8 a + 8\right)\cdot 13 + \left(7 a^{2} + 11 a + 11\right)\cdot 13^{2} + \left(10 a^{2} + a + 9\right)\cdot 13^{3} + \left(10 a^{2} + a + 2\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 9 a^{2} + 9 a + 9 + \left(6 a^{2} + a + 9\right)\cdot 13 + \left(12 a^{2} + 2 a + 3\right)\cdot 13^{2} + \left(a^{2} + 3 a + 8\right)\cdot 13^{3} + \left(12 a^{2} + a + 6\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 11 a^{2} + 11 a + 10 + \left(11 a^{2} + 10 a + 9\right)\cdot 13 + \left(4 a^{2} + 8\right)\cdot 13^{2} + \left(5 a^{2} + 8 a + 11\right)\cdot 13^{3} + \left(7 a^{2} + 9 a + 6\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 5 a^{2} + 2 a + 8 + \left(6 a^{2} + a\right)\cdot 13 + \left(11 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(2 a^{2} + a\right)\cdot 13^{3} + \left(3 a^{2} + 5 a + 12\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 11 a^{2} + 8 a + 6 + \left(12 a^{2} + 9 a + 9\right)\cdot 13 + \left(a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(7 a^{2} + 12\right)\cdot 13^{3} + \left(8 a^{2} + 2 a + 10\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 9 a^{2} + a + 3 + \left(12 a^{2} + 7 a + 2\right)\cdot 13 + 12\cdot 13^{2} + \left(10 a^{2} + 3 a + 4\right)\cdot 13^{3} + \left(7 a^{2} + 2 a + 7\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.