Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x + 3 x^{2} )^{4}$ |
| $1 - 8 x + 36 x^{2} - 104 x^{3} + 214 x^{4} - 312 x^{5} + 324 x^{6} - 216 x^{7} + 81 x^{8}$ | |
| Frobenius angles: | $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$ |
| Angle rank: | $1$ (numerical) |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $20736$ | $2085136$ | $84934656$ | $3429742096$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-4$ | $18$ | $68$ | $138$ | $236$ | $546$ | $1844$ | $6426$ | $20444$ | $60978$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3}$.
Endomorphism algebra over $\F_{3}$| The isogeny class factors as 1.3.ac 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.