Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 7 x^{2} + 25 x^{4}$ |
| Frobenius angles: | $\pm0.373408344447$, $\pm0.626591655553$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{3}, \sqrt{-17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 4 |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $33$ | $1089$ | $15444$ | $393129$ | $9761433$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $40$ | $126$ | $628$ | $3126$ | $15262$ | $78126$ | $393124$ | $1953126$ | $9757240$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2=x^6+x^3+2$
- $y^2=4 x^6+4 x^5+4 x^3+4 x^2+2$
- $y^2=3 x^6+3 x^5+3 x^3+3 x^2+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{3}, \sqrt{-17})\). |
| The base change of $A$ to $\F_{5^{2}}$ is 1.25.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
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 |
|---|---|---|
| 2.5.a_ah | $4$ | 2.625.c_bwd |
| 2.5.ad_i | $12$ | (not in LMFDB) |
| 2.5.d_i | $12$ | (not in LMFDB) |