# Properties

 Label 3.16.ar_fb_ayi Base Field $\F_{2^{4}}$ Dimension $3$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 9 x + 43 x^{2} - 144 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.108303609292$, $\pm0.441636942625$ Angle rank: $1$ (numerical)

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 $A(\F_{q^r})$ 1323 14982975 66603519108 277216037275275 1149181225223419773 4722152053598405571600 19344427927754309658188127 79227701795450409038033800275 324516077773744637635536636725148 1329227251536537493926963729830499375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 230 3969 64538 1045170 16776455 268457868 4294942322 68718952449 1099511012150

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.aj_br and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.16.ai : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.aj_br : $$\Q(\sqrt{-3}, \sqrt{37})$$.
Endomorphism algebra over $\overline{\F}_{2^{4}}$
 The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc $\times$ 1.16777216.fmx 2 . The endomorphism algebra for each factor is: 1.16777216.amdc : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16777216.fmx 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-111})$$$)$
All geometric endomorphisms are defined over $\F_{2^{24}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{8}}$  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 2.256.f_aix. The endomorphism algebra for each factor is: 1.256.abg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.256.f_aix : $$\Q(\sqrt{-3}, \sqrt{37})$$.
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 2.4096.a_fmx. The endomorphism algebra for each factor is: 1.4096.aey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.4096.a_fmx : $$\Q(\sqrt{-3}, \sqrt{37})$$.

## 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 3.16.ab_an_ce $2$ (not in LMFDB) 3.16.b_an_ace $2$ (not in LMFDB) 3.16.r_fb_yi $2$ (not in LMFDB) 3.16.ai_l_bo $3$ (not in LMFDB) 3.16.af_x_aem $3$ (not in LMFDB) 3.16.e_l_au $3$ (not in LMFDB) 3.16.n_dr_rs $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ab_an_ce $2$ (not in LMFDB) 3.16.b_an_ace $2$ (not in LMFDB) 3.16.r_fb_yi $2$ (not in LMFDB) 3.16.ai_l_bo $3$ (not in LMFDB) 3.16.af_x_aem $3$ (not in LMFDB) 3.16.e_l_au $3$ (not in LMFDB) 3.16.n_dr_rs $3$ (not in LMFDB) 3.16.aj_ch_alc $4$ (not in LMFDB) 3.16.j_ch_lc $4$ (not in LMFDB) 3.16.an_dr_ars $6$ (not in LMFDB) 3.16.ae_l_u $6$ (not in LMFDB) 3.16.f_x_em $6$ (not in LMFDB) 3.16.i_l_abo $6$ (not in LMFDB) 3.16.ai_v_abo $12$ (not in LMFDB) 3.16.ae_v_au $12$ (not in LMFDB) 3.16.a_l_a $12$ (not in LMFDB) 3.16.a_v_a $12$ (not in LMFDB) 3.16.e_v_u $12$ (not in LMFDB) 3.16.i_v_bo $12$ (not in LMFDB)