# Properties

 Label 3.2.af_n_aw 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: $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.456881978294$ 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 $A(\F_{q^r})$ 1 95 988 4275 39401 375440 2491763 15736275 125716084 1141643975

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 6 13 18 38 87 152 242 481 1086

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 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 $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.ey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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.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.e $\times$ 2.8.a_l. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.l 2 . The endomorphism algebra for each factor is: 1.64.a : $$\Q(\sqrt{-1})$$. 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$

## 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.ab_b_ac $2$ 3.4.b_b_i 3.2.b_b_c $2$ 3.4.b_b_i 3.2.f_n_w $2$ 3.4.b_b_i 3.2.ac_b_c $3$ 3.8.e_t_bs
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.2.ab_b_ac $2$ 3.4.b_b_i 3.2.b_b_c $2$ 3.4.b_b_i 3.2.f_n_w $2$ 3.4.b_b_i 3.2.ac_b_c $3$ 3.8.e_t_bs 3.2.f_n_w $4$ 3.16.b_ah_bo 3.2.c_b_ac $6$ (not in LMFDB) 3.2.ad_h_am $8$ (not in LMFDB) 3.2.d_h_m $8$ (not in LMFDB) 3.2.ac_d_ac $12$ (not in LMFDB) 3.2.c_d_c $12$ (not in LMFDB) 3.2.a_b_a $24$ (not in LMFDB) 3.2.a_d_a $24$ (not in LMFDB)