Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - x + 2 x^{2} - 4 x^{3} + 16 x^{4} - 20 x^{5} + 50 x^{6} - 125 x^{7} + 625 x^{8}$ |
| Frobenius angles: | $\pm0.135164221778$, $\pm0.344590989425$, $\pm0.604556650005$, $\pm0.813983417652$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 8.0.260758711400625.1 |
| Galois group: | $S_4\times C_2$ |
This isogeny class is simple and geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $544$ | $459136$ | $231835936$ | $165180603904$ | $94993541893024$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $29$ | $119$ | $673$ | $3115$ | $15725$ | $77782$ | $393873$ | $1959923$ | $9757649$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$| The endomorphism algebra of this simple isogeny class is 8.0.260758711400625.1. |
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 |
|---|---|---|
| 4.5.b_c_e_q | $2$ | (not in LMFDB) |