Properties

Label 2.3.a_af
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:  $1 - 5 x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.0932147493387$, $\pm0.906785250661$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{11})\)
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
$A(\F_{q^r})$ 5 25 740 5625 59525 547600 4785485 44555625 387399620 3543225625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 0 28 68 244 750 2188 6788 19684 60000

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{11})\).
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ac_h$4$2.81.ao_id
2.3.a_f$4$2.81.ao_id
2.3.c_h$4$2.81.ao_id
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ac_h$4$2.81.ao_id
2.3.a_f$4$2.81.ao_id
2.3.c_h$4$2.81.ao_id
2.3.ab_ac$12$(not in LMFDB)
2.3.b_ac$12$(not in LMFDB)