Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x + 8 x^{2} - 10 x^{3} + 25 x^{4} )^{2}$ |
| $1 - 4 x + 20 x^{2} - 52 x^{3} + 154 x^{4} - 260 x^{5} + 500 x^{6} - 500 x^{7} + 625 x^{8}$ | |
| Frobenius angles: | $\pm0.290805673663$, $\pm0.290805673663$, $\pm0.552340175673$, $\pm0.552340175673$ |
| Angle rank: | $2$ (numerical) |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 11$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $484$ | $1024144$ | $286218724$ | $154178734336$ | $101597166929764$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $50$ | $146$ | $634$ | $3322$ | $15554$ | $75994$ | $388698$ | $1960706$ | $9776690$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$| The isogeny class factors as 2.5.ac_i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-16 -2 \sqrt{3}})\)$)$ |
Base change
This is a primitive isogeny class.