# Properties

 Label 2.3.ae_i 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

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.0540867239847$, $\pm0.445913276015$ Angle rank: $1$ (numerical) Number field: $$\Q(\zeta_{8})$$ Galois group: $C_2^2$ Jacobians: 1

This isogeny class is simple but not geometrically simple.

## 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:

• $y^2=2x^6+x^5+x^3+x^2+x+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $68$ $626$ $4624$ $49282$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $10$ $24$ $54$ $200$ $730$ $2240$ $6494$ $19392$ $59050$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{8})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 2.9.a_ao and its endomorphism algebra is $$\Q(\zeta_{8})$$.

## 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.e_i$2$2.9.a_ao
2.3.ae_k$8$(not in LMFDB)
2.3.a_ac$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.e_i$2$2.9.a_ao
2.3.ae_k$8$(not in LMFDB)
2.3.a_ac$8$(not in LMFDB)
2.3.a_c$8$(not in LMFDB)
2.3.e_k$8$(not in LMFDB)
2.3.ac_b$24$(not in LMFDB)
2.3.c_b$24$(not in LMFDB)