Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )^{2}$ |
| $1 - 6 x + 19 x^{2} - 42 x^{3} + 69 x^{4} - 84 x^{5} + 76 x^{6} - 48 x^{7} + 16 x^{8}$ | |
| Frobenius angles: | $\pm0.123548644961$, $\pm0.123548644961$, $\pm0.456881978294$, $\pm0.456881978294$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $0$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1$ | $361$ | $5776$ | $29241$ | $923521$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $7$ | $9$ | $3$ | $27$ | $109$ | $207$ | $291$ | $513$ | $1147$ |
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 $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2}$| The isogeny class factors as 2.2.ad_f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$ |
| The base change of $A$ to $\F_{2^{6}}$ is 1.64.l 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$ |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 2.4.b_ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$ - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 2.8.a_l 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
Base change
This is a primitive isogeny class.