# Properties

 Label 2.3.ae_i Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ 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 Jacobians of 1 curves, and hence is principally polarizable:

• $y^2=2x^6+x^5+x^3+x^2+x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2 68 626 4624 49282 532100 4898098 42614784 381715394 3486898628

 $r$ 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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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)