Basic invariants
| Dimension: | $2$ |
| Group: | $D_{9}$ |
| Conductor: | \(2951\)\(\medspace = 13 \cdot 227 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.75836247976801.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{9}$ |
| Parity: | odd |
| Determinant: | 1.2951.2t1.a.a |
| Projective image: | $D_9$ |
| Projective stem field: | Galois closure of 9.1.75836247976801.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 4x^{8} + 9x^{7} - 37x^{6} + 187x^{5} - 499x^{4} + 843x^{3} - 734x^{2} + 546x - 13 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{3} + x + 35 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a^{2} + 37 a + 9 + \left(11 a^{2} + 5 a + 38\right)\cdot 41 + \left(16 a + 15\right)\cdot 41^{2} + \left(10 a^{2} + 19 a + 21\right)\cdot 41^{3} + \left(16 a^{2} + 26 a + 3\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 20 a^{2} + 38 a + 33 + \left(12 a^{2} + 30 a + 15\right)\cdot 41 + \left(34 a^{2} + 12 a + 39\right)\cdot 41^{2} + \left(18 a^{2} + 26 a + 22\right)\cdot 41^{3} + \left(30 a^{2} + 28 a + 11\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 15 a^{2} + 4 a + 3 + \left(10 a^{2} + 10 a + 37\right)\cdot 41 + \left(26 a^{2} + 32 a + 39\right)\cdot 41^{2} + \left(40 a^{2} + 30 a + 1\right)\cdot 41^{3} + \left(20 a^{2} + 16 a + 30\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 25 a^{2} + 29 a + 19 + \left(13 a^{2} + 25 a + 12\right)\cdot 41 + \left(14 a^{2} + 29 a + 25\right)\cdot 41^{2} + \left(9 a^{2} + 37 a + 34\right)\cdot 41^{3} + \left(4 a^{2} + 7 a + 22\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 4 a^{2} + 19 a + 36 + \left(14 a^{2} + 20 a + 16\right)\cdot 41 + \left(40 a^{2} + 23 a + 2\right)\cdot 41^{2} + \left(8 a^{2} + 13 a + 30\right)\cdot 41^{3} + \left(2 a^{2} + 6 a + 33\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 34 a^{2} + 39 a + 2 + \left(35 a^{2} + 3 a + 13\right)\cdot 41 + \left(35 a^{2} + 15 a + 5\right)\cdot 41^{2} + \left(36 a^{2} + 10 a + 13\right)\cdot 41^{3} + \left(34 a^{2} + 23 a + 39\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 33 a^{2} + 39 a + 15 + \left(35 a^{2} + 26 a + 40\right)\cdot 41 + \left(19 a^{2} + 34 a + 21\right)\cdot 41^{2} + \left(4 a^{2} + 40 a + 32\right)\cdot 41^{3} + \left(26 a^{2} + 19\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 17 a^{2} + 25 a + 31 + \left(14 a^{2} + 30 a + 30\right)\cdot 41 + \left(7 a^{2} + 4 a + 7\right)\cdot 41^{2} + \left(13 a^{2} + a + 19\right)\cdot 41^{3} + \left(8 a^{2} + 6 a + 10\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 9 }$ |
$=$ |
\( 6 a^{2} + 16 a + 20 + \left(16 a^{2} + 9 a\right)\cdot 41 + \left(26 a^{2} + 36 a + 6\right)\cdot 41^{2} + \left(21 a^{2} + 24 a + 29\right)\cdot 41^{3} + \left(20 a^{2} + 6 a + 33\right)\cdot 41^{4} +O(41^{5})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(3,6)(4,5)(8,9)$ | $0$ |
| $2$ | $3$ | $(1,4,9)(2,8,5)(3,7,6)$ | $-1$ |
| $2$ | $9$ | $(1,5,7,4,2,6,9,8,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,7,2,9,3,5,4,6,8)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,2,3,4,8,7,9,5,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.