Invariants
| Base field: | $\F_{397}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 34 x + 397 x^{2}$ |
| Frobenius angles: | $\pm0.174655262871$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $14$ |
| Isomorphism classes: | 14 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $364$ | $157248$ | $62571964$ | $24840781056$ | $9861722751244$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $364$ | $157248$ | $62571964$ | $24840781056$ | $9861722751244$ | $3915101757542976$ | $1554295350533798236$ | $617055253420108059648$ | $244970935601304037979308$ | $97253461433791920286306368$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which 0 are hyperelliptic):
- $y^2=x^3+170 x+340$
- $y^2=x^3+171 x+342$
- $y^2=x^3+34 x+68$
- $y^2=x^3+363 x+363$
- $y^2=x^3+91 x+91$
- $y^2=x^3+152 x+304$
- $y^2=x^3+146 x+292$
- $y^2=x^3+344 x+291$
- $y^2=x^3+396 x+395$
- $y^2=x^3+19 x+38$
- $y^2=x^3+2$
- $y^2=x^3+316 x+235$
- $y^2=x^3+49 x+98$
- $y^2=x^3+128 x+256$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{397}$.
Endomorphism algebra over $\F_{397}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
Base change
This is a primitive isogeny class.