Invariants
| Base field: | $\F_{439}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 439 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-439}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $30$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $440$ | $193600$ | $84604520$ | $37140998400$ | $16305067506200$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $440$ | $193600$ | $84604520$ | $37140998400$ | $16305067506200$ | $7157924804430400$ | $3142328914862177480$ | $1379482393550213145600$ | $605592770801153705930360$ | $265855226381739087038440000$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 30 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{439^{2}}$.
Endomorphism algebra over $\F_{439}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-439}) \). |
| The base change of $A$ to $\F_{439^{2}}$ is the simple isogeny class 1.192721.bhu and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $439$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.