# Properties

 Label 3.2.ae_l_as Base field $\F_{2}$ Dimension $3$ $p$-rank $2$ 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: $( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )^{2}$ Frobenius angles: $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.384973271919$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

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

## 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 $A(\F_{q^r})$ 4 320 2548 6400 19844 203840 2278532 18662400 129063844 960449600

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 11 23 23 19 47 139 287 491 911

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 1.2.ab 2 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 : $$\Q(\sqrt{-1})$$. 1.2.ab 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab 2 $\times$ 1.16.i. The endomorphism algebra for each factor is: 1.16.ab 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.16.i : 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 $\times$ 1.4.d 2 . The endomorphism algebra for each factor is: 1.4.a : $$\Q(\sqrt{-1})$$. 1.4.d 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$

## 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.ac_f_ag $2$ 3.4.g_v_bw 3.2.a_d_ac $2$ 3.4.g_v_bw 3.2.a_d_c $2$ 3.4.g_v_bw 3.2.c_f_g $2$ 3.4.g_v_bw 3.2.e_l_s $2$ 3.4.g_v_bw 3.2.ab_ab_g $3$ 3.8.o_dl_mm
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.2.ac_f_ag $2$ 3.4.g_v_bw 3.2.a_d_ac $2$ 3.4.g_v_bw 3.2.a_d_c $2$ 3.4.g_v_bw 3.2.c_f_g $2$ 3.4.g_v_bw 3.2.e_l_s $2$ 3.4.g_v_bw 3.2.ab_ab_g $3$ 3.8.o_dl_mm 3.2.ac_ab_g $4$ 3.16.g_bh_hs 3.2.c_ab_ag $4$ 3.16.g_bh_hs 3.2.e_l_s $4$ 3.16.g_bh_hs 3.2.ad_d_ac $6$ (not in LMFDB) 3.2.b_ab_ag $6$ (not in LMFDB) 3.2.d_d_c $6$ (not in LMFDB) 3.2.ac_h_ai $8$ (not in LMFDB) 3.2.a_ab_a $8$ (not in LMFDB) 3.2.a_f_a $8$ (not in LMFDB) 3.2.c_h_i $8$ (not in LMFDB) 3.2.ab_b_ae $24$ (not in LMFDB) 3.2.b_b_e $24$ (not in LMFDB)