# Properties

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

# Related objects

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.0292466093486$, $\pm0.637420057318$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Galois group: $C_2^2$ Jacobians: 1

This isogeny class is simple but not geometrically 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:

• $y^2=2x^6+2x^5+2x^4+2x^3+x+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $57$ $324$ $5529$ $58323$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $8$ $8$ $68$ $242$ $638$ $2102$ $6596$ $19304$ $58568$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{3}}$.

## 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.c_b$2$2.9.ac_af
2.3.e_k$3$2.27.au_fy
2.3.ae_k$6$2.729.ado_fhm
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.c_b$2$2.9.ac_af
2.3.e_k$3$2.27.au_fy
2.3.ae_k$6$2.729.ado_fhm
2.3.a_c$6$2.729.ado_fhm
2.3.a_ac$12$(not in LMFDB)
2.3.ae_i$24$(not in LMFDB)
2.3.e_i$24$(not in LMFDB)