Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | \(10003\)\(\medspace = 7 \cdot 1429 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of \(\Q(\sqrt{-10003}) \) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 15\cdot 23 + 4\cdot 23^{2} + 6\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 7\cdot 23 + 18\cdot 23^{2} + 16\cdot 23^{3} + 12\cdot 23^{4} +O(23^{5})\)
|
Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 } $ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)$ | $-1$ |