Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(10003\)\(\medspace = 7 \cdot 1429 \) |
Artin number field: | Galois closure of 3.3.100060009.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
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 $\Q_{ 17 }$ to precision 7.
Roots:
$r_{ 1 }$ | $=$ |
\( 7\cdot 17 + 10\cdot 17^{2} + 13\cdot 17^{3} + 13\cdot 17^{4} + 16\cdot 17^{5} + 14\cdot 17^{6} +O(17^{7})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 12\cdot 17 + 15\cdot 17^{2} + 14\cdot 17^{3} + 12\cdot 17^{4} + 11\cdot 17^{5} + 6\cdot 17^{6} +O(17^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 16 + 14\cdot 17 + 7\cdot 17^{2} + 5\cdot 17^{3} + 7\cdot 17^{4} + 5\cdot 17^{5} + 12\cdot 17^{6} +O(17^{7})\)
| |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $3$ | $(1,2,3)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |