Properties

Label 2.3.b_ac
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:  $1 + x - 2 x^{2} + 3 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.259881416005$, $\pm0.926548082672$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-11})\)
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
$A(\F_{q^r})$ 12 48 1296 7104 67332 518400 4606068 42311424 382124304 3514999728

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 5 44 89 275 710 2105 6449 19412 59525

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{3}}$.

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.ab_ac$2$2.9.af_q
2.3.ac_h$3$2.27.q_eo
2.3.a_f$6$2.729.au_chy
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ab_ac$2$2.9.af_q
2.3.ac_h$3$2.27.q_eo
2.3.a_f$6$2.729.au_chy
2.3.c_h$6$2.729.au_chy
2.3.a_af$12$(not in LMFDB)