Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 19 x^{2} )^{2}$ |
$1 - 14 x + 87 x^{2} - 266 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.203259864187$, $\pm0.203259864187$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $169$ | $123201$ | $47831056$ | $17140831929$ | $6146645853049$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $340$ | $6972$ | $131524$ | $2482386$ | $47067046$ | $893900454$ | $16983361924$ | $322685744388$ | $6131056405300$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+7x^5+6x^4+13x^3+6x^2+7x+10$
- $y^2=2x^6+2x^3+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ah 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.