Properties

Label 2.3.ab_g
Base field $\F_{3}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$
  $1 - x + 6 x^{2} - 3 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.406785250661$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12$ $240$ $1008$ $4800$ $51972$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $21$ $36$ $57$ $213$ $774$ $2271$ $6513$ $19548$ $59061$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ab $\times$ 1.3.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.f $\times$ 1.9.g. The endomorphism algebra for each factor is:
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.b_g$2$2.9.l_bw
2.3.ae_j$3$2.27.i_cc
2.3.c_d$3$2.27.i_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.b_g$2$2.9.l_bw
2.3.ae_j$3$2.27.i_cc
2.3.c_d$3$2.27.i_cc
2.3.ae_j$6$2.729.bs_bji
2.3.ac_d$6$2.729.bs_bji
2.3.c_d$6$2.729.bs_bji
2.3.e_j$6$2.729.bs_bji