Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 16 x^{2} )^{2}$ |
$1 - 14 x + 81 x^{2} - 224 x^{3} + 256 x^{4}$ | |
Frobenius angles: | $\pm0.160861246510$, $\pm0.160861246510$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $100$ | $57600$ | $16728100$ | $4324377600$ | $1103025062500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $223$ | $4083$ | $65983$ | $1051923$ | $16793503$ | $268495923$ | $4295129983$ | $68719648083$ | $1099510224223$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=x^5+a^3x^4+x^3+(a^3+1)x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$The isogeny class factors as 1.16.ah 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This is a primitive isogeny class.