Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 19 x^{2} )^{2}$ |
$1 - 16 x + 102 x^{2} - 304 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.130073469147$, $\pm0.130073469147$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $1$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $144$ | $112896$ | $46294416$ | $16995815424$ | $6138164811024$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $310$ | $6748$ | $130414$ | $2478964$ | $47067046$ | $893986636$ | $16984080094$ | $322689651172$ | $6131072060950$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.