# Properties

 Label 3.2.ae_j_ao 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: $( 1 - 2 x + 2 x^{2} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.174442860055$, $\pm0.250000000000$, $\pm0.546783656212$ Angle rank: $2$ (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 $A(\F_{q^r})$ 2 140 806 5600 72242 394940 1813198 14716800 134765618 1036672700

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 7 11 23 59 91 111 223 515 987

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 2.2.ac_d 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 $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.ac_b. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ac_b : 4.0.1088.2.
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 $\times$ 2.4.c_b. 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.a_b_ac $2$ 3.4.c_f_q 3.2.a_b_c $2$ 3.4.c_f_q 3.2.e_j_o $2$ 3.4.c_f_q
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.2.a_b_ac $2$ 3.4.c_f_q 3.2.a_b_c $2$ 3.4.c_f_q 3.2.e_j_o $2$ 3.4.c_f_q 3.2.ac_f_ai $8$ (not in LMFDB) 3.2.c_f_i $8$ (not in LMFDB)