Properties

Label 2.1215.9t3.a
Dimension $2$
Group $D_{9}$
Conductor $1215$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{9}$
Conductor:\(1215\)\(\medspace = 3^{5} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 9.1.242137805625.3
Galois orbit size: $3$
Smallest permutation container: $D_{9}$
Parity: odd
Projective image: $D_9$
Projective field: Galois closure of 9.1.242137805625.3

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{3} + 3x + 42 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 31 a^{2} + 41 a + 15 + \left(33 a^{2} + 25 a + 20\right)\cdot 47 + \left(44 a^{2} + 7 a + 42\right)\cdot 47^{2} + \left(5 a^{2} + 11 a + 11\right)\cdot 47^{3} + \left(25 a^{2} + 35 a + 3\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 a^{2} + 39 a + 28 + \left(36 a^{2} + 6 a + 25\right)\cdot 47 + \left(35 a^{2} + 24 a + 24\right)\cdot 47^{2} + \left(38 a^{2} + 9 a + 30\right)\cdot 47^{3} + \left(14 a^{2} + 12 a + 29\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 a^{2} + 43 a + 12 + \left(42 a^{2} + 12 a + 37\right)\cdot 47 + \left(3 a^{2} + 3 a + 7\right)\cdot 47^{2} + \left(20 a^{2} + 14 a + 40\right)\cdot 47^{3} + 40 a\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 a^{2} + 40 a + 26 + \left(27 a^{2} + 14 a + 7\right)\cdot 47 + \left(24 a^{2} + 27 a + 2\right)\cdot 47^{2} + \left(2 a^{2} + 34 a + 5\right)\cdot 47^{3} + \left(33 a^{2} + 9 a + 19\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 15 a^{2} + 5 a + 30 + \left(26 a^{2} + 19 a + 5\right)\cdot 47 + \left(29 a^{2} + 21 a + 12\right)\cdot 47^{2} + \left(30 a^{2} + 44 a + 14\right)\cdot 47^{3} + \left(15 a^{2} + a + 31\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 a^{2} + 11 a + 9 + \left(24 a^{2} + 19 a + 2\right)\cdot 47 + \left(18 a^{2} + 16 a + 37\right)\cdot 47^{2} + \left(24 a^{2} + 45 a + 1\right)\cdot 47^{3} + \left(13 a^{2} + 43 a + 27\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 2 a^{2} + 14 a + 4 + \left(24 a^{2} + 14 a + 1\right)\cdot 47 + \left(13 a^{2} + 15 a + 27\right)\cdot 47^{2} + \left(2 a^{2} + 26 a + 4\right)\cdot 47^{3} + \left(7 a^{2} + 46 a + 14\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 31 a^{2} + 30 a + 15 + \left(10 a^{2} + 43 a + 21\right)\cdot 47 + \left(22 a^{2} + 6 a + 44\right)\cdot 47^{2} + \left(18 a^{2} + 28 a + 36\right)\cdot 47^{3} + \left(42 a^{2} + 11 a + 37\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( a^{2} + 12 a + 2 + \left(10 a^{2} + 31 a + 20\right)\cdot 47 + \left(42 a^{2} + 18 a + 37\right)\cdot 47^{2} + \left(44 a^{2} + 21 a + 42\right)\cdot 47^{3} + \left(35 a^{2} + 33 a + 24\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character values
$c1$ $c2$ $c3$
$1$ $1$ $()$ $2$ $2$ $2$
$9$ $2$ $(1,8)(2,5)(3,4)(7,9)$ $0$ $0$ $0$
$2$ $3$ $(1,7,2)(3,6,4)(5,9,8)$ $-1$ $-1$ $-1$
$2$ $9$ $(1,4,9,7,3,8,2,6,5)$ $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ $\zeta_{9}^{5} + \zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$
$2$ $9$ $(1,9,3,2,5,4,7,8,6)$ $\zeta_{9}^{5} + \zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$
$2$ $9$ $(1,3,5,7,6,9,2,4,8)$ $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ $\zeta_{9}^{5} + \zeta_{9}^{4}$
The blue line marks the conjugacy class containing complex conjugation.