Basic invariants
| Dimension: | $1$ |
| Group: | $C_9$ |
| Conductor: | \(37\) |
| Artin field: | Galois closure of 9.9.3512479453921.1 |
| Galois orbit size: | $6$ |
| Smallest permutation container: | $C_9$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{37}(7,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - 16x^{7} + 11x^{6} + 66x^{5} - 32x^{4} - 73x^{3} + 7x^{2} + 7x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a^{2} + 8 a + 2 + \left(5 a + 5\right)\cdot 11 + \left(6 a^{2} + 3 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 5 a + 2\right)\cdot 11^{3} + \left(10 a^{2} + 9 a + 5\right)\cdot 11^{4} + \left(3 a^{2} + 6\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a^{2} + 2 a + 6 + \left(9 a^{2} + a + 6\right)\cdot 11 + \left(a^{2} + a + 6\right)\cdot 11^{2} + \left(7 a^{2} + 10 a + 10\right)\cdot 11^{3} + \left(a^{2} + a + 3\right)\cdot 11^{4} + \left(6 a^{2} + 8 a + 9\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + \left(6 a^{2} + 5 a + 4\right)\cdot 11 + \left(9 a + 4\right)\cdot 11^{2} + \left(9 a^{2} + a + 4\right)\cdot 11^{3} + \left(9 a + 4\right)\cdot 11^{4} + \left(10 a^{2} + 3 a + 8\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( a^{2} + 2 a + 5 + \left(3 a^{2} + 7 a + 7\right)\cdot 11 + \left(7 a^{2} + 2 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(5 a^{2} + 4 a + 3\right)\cdot 11^{4} + \left(3 a^{2} + a + 3\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a^{2} + 4 a + 3 + \left(7 a^{2} + 5 a + 9\right)\cdot 11 + \left(3 a^{2} + a\right)\cdot 11^{2} + \left(2 a^{2} + a + 6\right)\cdot 11^{3} + \left(4 a^{2} + a\right)\cdot 11^{4} + \left(6 a^{2} + 2 a + 5\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 10 a + 6 + \left(a^{2} + 8 a + 9\right)\cdot 11 + \left(3 a^{2} + 9 a + 7\right)\cdot 11^{2} + \left(6 a^{2} + a\right)\cdot 11^{3} + \left(4 a^{2} + 8 a + 2\right)\cdot 11^{4} + \left(8 a^{2} + 5 a + 6\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 9 a^{2} + 6 a + \left(10 a^{2} + 5 a + 3\right)\cdot 11 + \left(5 a^{2} + a\right)\cdot 11^{2} + \left(3 a^{2} + 8 a + 4\right)\cdot 11^{3} + \left(7 a^{2} + 4 a + 8\right)\cdot 11^{4} + \left(5 a^{2} + 7 a + 7\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 10 a^{2} + a + 5 + \left(3 a^{2} + 8\right)\cdot 11 + \left(a^{2} + 8 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + a + 9\right)\cdot 11^{3} + \left(10 a^{2} + 5 a + 8\right)\cdot 11^{4} + \left(9 a^{2} + a + 9\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 9 }$ | $=$ |
\( 7 a^{2} + a + 7 + \left(4 a + 1\right)\cdot 11 + \left(3 a^{2} + 6 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + 6 a\right)\cdot 11^{3} + \left(9 a^{2} + 10 a + 7\right)\cdot 11^{4} + \left(a + 9\right)\cdot 11^{5} +O(11^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,2,9)(3,4,6)(5,8,7)$ | $-\zeta_{9}^{3} - 1$ |
| $1$ | $3$ | $(1,9,2)(3,6,4)(5,7,8)$ | $\zeta_{9}^{3}$ |
| $1$ | $9$ | $(1,6,8,2,3,7,9,4,5)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,8,3,9,5,6,2,7,4)$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ | $9$ | $(1,3,5,2,4,8,9,6,7)$ | $\zeta_{9}^{5}$ |
| $1$ | $9$ | $(1,7,6,9,8,4,2,5,3)$ | $\zeta_{9}^{4}$ |
| $1$ | $9$ | $(1,4,7,2,6,5,9,3,8)$ | $\zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,5,4,9,7,3,2,8,6)$ | $\zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.