Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 19 x^{2} )^{2}$ |
$1 - 12 x + 74 x^{2} - 228 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.258380448083$, $\pm0.258380448083$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $196$ | $132496$ | $48804196$ | $17171481600$ | $6140553384196$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $366$ | $7112$ | $131758$ | $2479928$ | $47041566$ | $893773112$ | $16983053278$ | $322686513128$ | $6131068835406$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=15x^6+9x^4+9x^2+15$
- $y^2=10x^6+14x^5+9x^4+11x^3+9x^2+14x+10$
- $y^2=18x^6+18x^4+x^3+x^2+3x$
- $y^2=15x^6+5x^5+8x^4+5x^3+14x^2+7x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$ |
Base change
This is a primitive isogeny class.