Invariants
Base field: | $\F_{3^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 31 x + 243 x^{2} )^{2}$ |
$1 - 62 x + 1447 x^{2} - 15066 x^{3} + 59049 x^{4}$ | |
Frobenius angles: | $\pm0.0339262533067$, $\pm0.0339262533067$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $45369$ | $3431030625$ | $205684817824656$ | $12156915630659765625$ | $717895300637162526680169$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $182$ | $58100$ | $14334524$ | $3486569348$ | $847285438082$ | $205891086040550$ | $50031544441963574$ | $12157665449879954948$ | $2954312706426007289732$ | $717897987690212974740500$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{5}}$.
Endomorphism algebra over $\F_{3^{5}}$The isogeny class factors as 1.243.abf 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{5}}$.
Subfield | Primitive Model |
$\F_{3}$ | 2.3.ac_h |