# Properties

 Label 3.2.ae_k_ar Base field $\F_{2}$ Dimension $3$ $p$-rank $3$ Ordinary Yes Supersingular No Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $3$ L-polynomial: $( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.123548644961$, $\pm0.384973271919$, $\pm0.456881978294$ Angle rank: $2$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2 152 1064 2736 21142 323456 3131242 20142432 135386552 1078157432

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 9 14 9 19 78 181 305 518 1029

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ab $\times$ 2.2.ad_f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.l 2 . The endomorphism algebra for each factor is: 1.64.aj : $$\Q(\sqrt{-7})$$. 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 2.4.b_ad. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.f $\times$ 2.8.a_l. The endomorphism algebra for each factor is:

## 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 3.2.ac_e_ah $2$ 3.4.e_e_ab 3.2.c_e_h $2$ 3.4.e_e_ab 3.2.e_k_r $2$ 3.4.e_e_ab 3.2.ab_b_b $3$ 3.8.f_t_cd
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.2.ac_e_ah $2$ 3.4.e_e_ab 3.2.c_e_h $2$ 3.4.e_e_ab 3.2.e_k_r $2$ 3.4.e_e_ab 3.2.ab_b_b $3$ 3.8.f_t_cd 3.2.b_b_ab $6$ (not in LMFDB) 3.2.ab_d_ab $12$ (not in LMFDB) 3.2.b_d_b $12$ (not in LMFDB)