# Properties

 Label 3.2.ae_k_aq Base field $\F_{2}$ Dimension $3$ $p$-rank $0$ Ordinary No Supersingular Yes Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $3$ L-polynomial: $( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}$ Frobenius angles: $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.5$ Angle rank: $0$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular.

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

## 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 $A(\F_{q^r})$ 3 225 1521 5625 55473 342225 1647201 11390625 118688193 1144130625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 9 17 25 49 81 97 161 449 1089

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 1.2.a 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.a : $$\Q(\sqrt{-2})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{8}}$.
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.e. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.4.e : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i 2 . The endomorphism algebra for each factor is: 1.16.ai : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.

## 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.a_c_a $2$ 3.4.e_m_bg 3.2.e_k_q $2$ 3.4.e_m_bg 3.2.c_e_i $3$ 3.8.i_bo_ey
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.2.a_c_a $2$ 3.4.e_m_bg 3.2.e_k_q $2$ 3.4.e_m_bg 3.2.c_e_i $3$ 3.8.i_bo_ey 3.2.ac_e_ai $6$ (not in LMFDB) 3.2.ag_s_abg $8$ (not in LMFDB) 3.2.ac_ac_i $8$ (not in LMFDB) 3.2.ac_c_a $8$ (not in LMFDB) 3.2.ac_g_ai $8$ (not in LMFDB) 3.2.a_ac_a $8$ (not in LMFDB) 3.2.a_g_a $8$ (not in LMFDB) 3.2.c_ac_ai $8$ (not in LMFDB) 3.2.c_c_a $8$ (not in LMFDB) 3.2.c_g_i $8$ (not in LMFDB) 3.2.g_s_bg $8$ (not in LMFDB) 3.2.ae_i_am $24$ (not in LMFDB) 3.2.ac_a_e $24$ (not in LMFDB) 3.2.ac_e_ae $24$ (not in LMFDB) 3.2.a_a_ae $24$ (not in LMFDB) 3.2.a_a_a $24$ (not in LMFDB) 3.2.a_a_e $24$ (not in LMFDB) 3.2.a_e_a $24$ (not in LMFDB) 3.2.c_a_ae $24$ (not in LMFDB) 3.2.c_e_e $24$ (not in LMFDB) 3.2.e_i_m $24$ (not in LMFDB)