Properties

Label 2.9.af_q
Base field $\F_{3^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 5 x + 16 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.146903834656$, $\pm0.519762832011$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:  $C_2^2$
Jacobians:  $11$
Isomorphism classes:  9

This isogeny class is simple but not geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $48$ $7104$ $518400$ $42311424$ $3514999728$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $89$ $710$ $6449$ $59525$ $534158$ $4785485$ $43046369$ $387462230$ $3486891929$

Jacobians and polarizations

This isogeny class contains the Jacobians of 11 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).
Endomorphism algebra over $\overline{\F}_{3^{2}}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

SubfieldPrimitive Model
$\F_{3}$2.3.ab_ac
$\F_{3}$2.3.b_ac

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.9.f_q$2$2.81.h_abg
2.9.k_br$3$2.729.au_chy
2.9.ak_br$6$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.9.f_q$2$2.81.h_abg
2.9.k_br$3$2.729.au_chy
2.9.ak_br$6$(not in LMFDB)
2.9.a_ah$6$(not in LMFDB)
2.9.a_h$12$(not in LMFDB)