# Properties

 Label 3.2.af_o_ay Base field $\F_{2}$ Dimension $3$ $p$-rank $1$ Ordinary No 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: $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}$ Frobenius angles: $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1/2, 1/2, 1]$

## 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 $A(\F_{q^r})$ 2 200 2366 10000 36982 236600 1813198 14580000 119844998 1017005000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 8 22 32 38 56 110 224 454 968

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 1.2.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.2.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.2.ab : $$\Q(\sqrt{-7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i 2 . The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 2 $\times$ 1.4.d. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.4.d : $$\Q(\sqrt{-7})$$.

## 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.ad_g_ai $2$ 3.4.d_m_y 3.2.ab_c_a $2$ 3.4.d_m_y 3.2.b_c_a $2$ 3.4.d_m_y 3.2.d_g_i $2$ 3.4.d_m_y 3.2.f_o_y $2$ 3.4.d_m_y 3.2.b_c_g $3$ 3.8.n_dc_lc
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.2.ad_g_ai $2$ 3.4.d_m_y 3.2.ab_c_a $2$ 3.4.d_m_y 3.2.b_c_a $2$ 3.4.d_m_y 3.2.d_g_i $2$ 3.4.d_m_y 3.2.f_o_y $2$ 3.4.d_m_y 3.2.b_c_g $3$ 3.8.n_dc_lc 3.2.ad_g_ai $4$ 3.16.p_ds_qa 3.2.ab_c_a $4$ 3.16.p_ds_qa 3.2.b_c_a $4$ 3.16.p_ds_qa 3.2.d_g_i $4$ 3.16.p_ds_qa 3.2.f_o_y $4$ 3.16.p_ds_qa 3.2.ad_g_ak $6$ (not in LMFDB) 3.2.ab_c_ag $6$ (not in LMFDB) 3.2.d_g_k $6$ (not in LMFDB) 3.2.ad_i_am $8$ (not in LMFDB) 3.2.ab_ac_e $8$ (not in LMFDB) 3.2.ab_e_ae $8$ (not in LMFDB) 3.2.ab_g_ae $8$ (not in LMFDB) 3.2.b_ac_ae $8$ (not in LMFDB) 3.2.b_e_e $8$ (not in LMFDB) 3.2.b_g_e $8$ (not in LMFDB) 3.2.d_i_m $8$ (not in LMFDB) 3.2.ad_g_ak $12$ (not in LMFDB) 3.2.ab_c_ag $12$ (not in LMFDB) 3.2.b_c_g $12$ (not in LMFDB) 3.2.d_g_k $12$ (not in LMFDB) 3.2.ab_a_c $24$ (not in LMFDB) 3.2.ab_e_ac $24$ (not in LMFDB) 3.2.b_a_ac $24$ (not in LMFDB) 3.2.b_e_c $24$ (not in LMFDB)