Properties

Label 2.3.ae_k
Base field $\F_{3}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
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:  $( 1 - 2 x + 3 x^{2} )^{2}$
  $1 - 4 x + 10 x^{2} - 12 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.304086723985$, $\pm0.304086723985$
Angle rank:  $1$ (numerical)
Jacobians:  1

This isogeny class is not simple, 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

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

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $144$ $1444$ $9216$ $58564$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $14$ $48$ $110$ $240$ $638$ $2016$ $6494$ $20064$ $60014$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
All geometric endomorphisms are defined over $\F_{3}$.

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.a_c$2$2.9.e_w
2.3.e_k$2$2.9.e_w
2.3.c_b$3$2.27.u_fy
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_c$2$2.9.e_w
2.3.e_k$2$2.9.e_w
2.3.c_b$3$2.27.u_fy
2.3.a_ac$4$2.81.bc_nu
2.3.ac_b$6$2.729.ado_fhm
2.3.ae_i$8$(not in LMFDB)
2.3.e_i$8$(not in LMFDB)