Invariants
| Base field: | $\F_{5^{4}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 8 x + 625 x^{2}$ |
| Frobenius angles: | $\pm0.551149423452$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-609}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 16 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $634$ | $391812$ | $244126138$ | $152587265280$ | $95367445698394$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $634$ | $391812$ | $244126138$ | $152587265280$ | $95367445698394$ | $59604645053769732$ | $37252902973606002778$ | $23283064365301081052160$ | $14551915228374422072503354$ | $9094947017729275493172530052$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{241} x+a^{242}$
- $y^2=x^3+a^{549} x+a^{550}$
- $y^2=x^3+a^{213} x+a^{213}$
- $y^2=x^3+a^{417} x+a^{417}$
- $y^2=x^3+a^{477} x+a^{477}$
- $y^2=x^3+a^{69} x+a^{69}$
- $y^2=x^3+a^{361} x+a^{362}$
- $y^2=x^3+a^{573} x+a^{574}$
- $y^2=x^3+a^{513} x+a^{513}$
- $y^2=x^3+a^{345} x+a^{345}$
- $y^2=x^3+a^{481} x+a^{481}$
- $y^2=x^3+a^{333} x+a^{333}$
- $y^2=x^3+a^{221} x+a^{221}$
- $y^2=x^3+a^{441} x+a^{441}$
- $y^2=x^3+a^{169} x+a^{169}$
- $y^2=x^3+a^{533} x+a^{533}$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5^{4}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-609}) \). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 1.625.ai | $2$ | (not in LMFDB) |