# 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

# Related objects

## 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:

• $y^2=2x^6+2x^4+2x^2+2$

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

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

## 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)