Properties

Label 2.3.ad_g
Base field $\F_{3}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 + 3 x^{2} )$
  $1 - 3 x + 6 x^{2} - 9 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.5$
Angle rank:  $0$ (numerical)
Jacobians:  1

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $112$ $784$ $5824$ $66124$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $13$ $28$ $73$ $271$ $838$ $2269$ $6481$ $19684$ $59293$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 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 $\F_{3^{6}}$ is 1.729.cc 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field

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.ag_p$3$2.27.a_cc
2.3.a_ad$3$2.27.a_cc
2.3.a_g$3$2.27.a_cc
2.3.d_g$3$2.27.a_cc
2.3.g_p$3$2.27.a_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ag_p$3$2.27.a_cc
2.3.a_ad$3$2.27.a_cc
2.3.a_g$3$2.27.a_cc
2.3.d_g$3$2.27.a_cc
2.3.g_p$3$2.27.a_cc
2.3.a_ad$6$2.729.ee_gmg
2.3.a_ag$12$(not in LMFDB)
2.3.a_d$12$(not in LMFDB)
2.3.a_a$24$(not in LMFDB)