Properties

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

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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.

Point counts of the abelian variety

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

Point counts of the (virtual) curve

$r$ 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.
TwistExtension DegreeCommon 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)