Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - 4 x + 4 x^{2} + 7 x^{3} - 21 x^{4} + 14 x^{5} + 16 x^{6} - 32 x^{7} + 16 x^{8}$ |
| Frobenius angles: | $\pm0.0764513550391$, $\pm0.143118021706$, $\pm0.323548644961$, $\pm0.943118021706$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{15})\) |
| Galois group: | $C_4\times C_2$ |
| Jacobians: | $0$ |
| Isomorphism classes: | 1 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1$ | $31$ | $7471$ | $55831$ | $1343281$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $-3$ | $14$ | $13$ | $39$ | $54$ | $146$ | $301$ | $608$ | $1147$ |
Jacobians and polarizations
This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{30}}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{15})\). |
| The base change of $A$ to $\F_{2^{30}}$ is 1.1073741824.acgor 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 the simple isogeny class 4.4.ai_be_acv_ft and its endomorphism algebra is \(\Q(\zeta_{15})\). - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 4.8.f_h_z_ez and its endomorphism algebra is \(\Q(\zeta_{15})\). - Endomorphism algebra over $\F_{2^{5}}$
The base change of $A$ to $\F_{2^{5}}$ is 2.32.d_bj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$ - Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 4.64.al_cf_cz_agqx and its endomorphism algebra is \(\Q(\zeta_{15})\). - Endomorphism algebra over $\F_{2^{10}}$
The base change of $A$ to $\F_{2^{10}}$ is 2.1024.cj_dzt 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$ - Endomorphism algebra over $\F_{2^{15}}$
The base change of $A$ to $\F_{2^{15}}$ is 2.32768.a_acgor 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
Base change
This is a primitive isogeny class.
Twists
Additional information
This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.