Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} )^{2}$ |
$1 - 6 x + 19 x^{2} - 30 x^{3} + 25 x^{4}$ | |
Frobenius angles: | $\pm0.265942140215$, $\pm0.265942140215$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9$ | $729$ | $20736$ | $455625$ | $10131489$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $28$ | $162$ | $724$ | $3240$ | $15478$ | $77112$ | $388324$ | $1951290$ | $9771628$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+4x^5+4x^4+2x^3+4x^2+4x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This is a primitive isogeny class.