Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x^{2} + 25 x^{4}$ |
| Frobenius angles: | $\pm0.184505059783$, $\pm0.815494940217$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-6}, \sqrt{14})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $22$ | $484$ | $15862$ | $435600$ | $9760102$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $18$ | $126$ | $694$ | $3126$ | $16098$ | $78126$ | $390814$ | $1953126$ | $9754578$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=3 x^6+4 x^5+3 x^3+x^2+4 x$
- $y^2=x^6+3 x^5+x^3+2 x^2+3 x$
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 \(\Q(\sqrt{-6}, \sqrt{14})\). |
| The base change of $A$ to $\F_{5^{2}}$ is 1.25.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$ |
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_e | $4$ | 2.625.cq_doo |