Properties

 Label 2.3.a_ae Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - 4 x^{2} + 9 x^{4}$ Frobenius angles: $\pm0.133860236401$, $\pm0.866139763599$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Galois group: $C_2^2$ Jacobians: 0

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 6 36 774 7056 59286 599076 4778934 45158400 387409446 3514829796

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 2 28 86 244 818 2188 6878 19684 59522

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-2}, \sqrt{-5})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{2}}$ is 1.9.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
All geometric endomorphisms are defined over $\F_{3^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.a_e $4$ 2.81.e_gk 2.3.ac_c $8$ (not in LMFDB) 2.3.c_c $8$ (not in LMFDB)