Properties

Label 2.3.a_e
Base Field $\F_{3}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $1 + 4 x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.366139763599$, $\pm0.633860236401$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{2}, \sqrt{-5})\)
Galois group:  $C_2^2$
Jacobians:  1

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 contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 14 196 686 7056 58814 470596 4787006 45158400 387431534 3459086596

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 18 28 86 244 642 2188 6878 19684 58578

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.e 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_ae$4$2.81.e_gk
2.3.ac_c$8$(not in LMFDB)
2.3.c_c$8$(not in LMFDB)