Basic invariants
| Dimension: | $1$ |
| Group: | $C_9$ |
| Conductor: | \(19\) |
| Artin number field: | Galois closure of \(\Q(\zeta_{19})^+\) |
| Galois orbit size: | $6$ |
| Smallest permutation container: | $C_9$ |
| Parity: | even |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{2} + 4 a + \left(a^{2} + 6 a + 2\right)\cdot 11 + \left(4 a^{2} + 3 a + 4\right)\cdot 11^{2} + \left(a^{2} + 3 a + 5\right)\cdot 11^{3} + \left(4 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(8 a^{2} + 8 a + 6\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 2 }$ |
$=$ |
\( 4 a^{2} + 2 a + 8 + \left(2 a^{2} + 4 a + 9\right)\cdot 11 + \left(5 a^{2} + 9 a + 9\right)\cdot 11^{2} + \left(9 a^{2} + a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 2 a + 9\right)\cdot 11^{4} + \left(9 a^{2} + 6 a + 9\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 2 a^{2} + 8 a + 9 + \left(7 a^{2} + 5 a + 8\right)\cdot 11 + \left(a^{2} + 8 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 4 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 4 a + 7\right)\cdot 11^{4} + \left(2 a^{2} + 7 a + 7\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 9 a^{2} + 6 a + 8 + \left(a^{2} + 2 a + 2\right)\cdot 11 + \left(5 a^{2} + 9\right)\cdot 11^{2} + \left(6 a^{2} + 3 a + 4\right)\cdot 11^{3} + \left(6 a^{2} + 9 a + 6\right)\cdot 11^{4} + \left(2 a^{2} + 8 a + 6\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a^{2} + a + 2 + \left(a^{2} + a + 1\right)\cdot 11 + \left(4 a^{2} + 4 a + 1\right)\cdot 11^{2} + \left(3 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 10 a^{2} + a + 2 + \left(7 a^{2} + 2 a + 7\right)\cdot 11 + \left(a^{2} + 7 a + 4\right)\cdot 11^{2} + \left(3 a^{2} + 4 a\right)\cdot 11^{3} + \left(3 a + 9\right)\cdot 11^{4} + \left(4 a + 6\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 7 a^{2} + \left(7 a^{2} + 8 a + 7\right)\cdot 11 + \left(4 a^{2} + 2 a + 4\right)\cdot 11^{2} + \left(5 a^{2} + 9 a + 2\right)\cdot 11^{3} + \left(6 a^{2} + a + 1\right)\cdot 11^{4} + \left(5 a^{2} + 3 a\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 8 }$ |
$=$ |
\( 7 a^{2} + 10 a + \left(6 a^{2} + 8 a + 2\right)\cdot 11 + \left(a^{2} + 9 a + 4\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} + \left(10 a^{2} + 10 a + 9\right)\cdot 11^{4} + \left(8 a^{2} + a\right)\cdot 11^{5} +O(11^{6})\)
| $r_{ 9 }$ |
$=$ |
\( 8 a^{2} + a + 5 + \left(7 a^{2} + 5 a + 3\right)\cdot 11 + \left(4 a^{2} + 9 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + 7 a + 9\right)\cdot 11^{3} + \left(5 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(7 a^{2} + 5 a + 9\right)\cdot 11^{5} +O(11^{6})\)
| |
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 values | |||||
| $c1$ | $c2$ | $c3$ | $c4$ | $c5$ | $c6$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)(7,8,9)$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)(7,9,8)$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ | $-\zeta_{9}^{3} - 1$ | $\zeta_{9}^{3}$ |
| $1$ | $9$ | $(1,2,8,6,3,9,4,5,7)$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,8,3,4,7,2,6,9,5)$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $1$ | $9$ | $(1,3,7,6,5,8,4,2,9)$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ |
| $1$ | $9$ | $(1,9,2,4,8,5,6,7,3)$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ |
| $1$ | $9$ | $(1,5,9,6,2,7,4,3,8)$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ |
| $1$ | $9$ | $(1,7,5,4,9,3,6,8,2)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ |