Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 19 x^{2} )( 1 - 7 x + 19 x^{2} )$ |
$1 - 15 x + 94 x^{2} - 285 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.130073469147$, $\pm0.203259864187$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 4 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $156$ | $117936$ | $47056464$ | $17068169664$ | $6142403868276$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $325$ | $6860$ | $130969$ | $2480675$ | $47067046$ | $893943545$ | $16983721009$ | $322687697780$ | $6131064233125$ |
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 $\F_{19^{6}}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ai $\times$ 1.19.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{19^{6}}$ is 1.47045881.pra 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{19^{2}}$
The base change of $A$ to $\F_{19^{2}}$ is 1.361.aba $\times$ 1.361.al. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{19^{3}}$
The base change of $A$ to $\F_{19^{3}}$ is 1.6859.ace $\times$ 1.6859.ce. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.