Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 3 x + 8 x^{2} - 15 x^{3} + 25 x^{4}$ |
| Frobenius angles: | $\pm0.206741677780$, $\pm0.540075011113$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $832$ | $15808$ | $389376$ | $10332496$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $33$ | $126$ | $625$ | $3303$ | $15990$ | $77787$ | $389377$ | $1953126$ | $9761433$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+x^5+4x^4+x^3+4x+4$
- $y^2=3x^6+2x^5+4x^4+3x^3+4x^2+x+3$
- $y^2=3x^5+2x^4+4x^2+3x+2$
- $y^2=3x^6+2x^5+4x^3+4x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{6}}$.
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 $\F_{5^{6}}$ is 1.15625.ha 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 2.25.h_y and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\). - Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is the simple isogeny class 2.125.a_ha and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).
Base change
This is a primitive isogeny class.