# Properties

 Label 3.16.au_gx_abjk 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: $( 1 - 4 x )^{2}( 1 - 7 x + 16 x^{2} )( 1 - 5 x + 16 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.160861246510$, $\pm0.285098958592$ Angle rank: $2$ (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 $A(\F_{q^r})$ 1080 14256000 68374280520 282218904000000 1153030973368527000 4720935756600279216000 19340269453314591891183720 79225708645026070037856000000 324516991224084672707130388868280 1329226874624178336332923025310000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 215 4077 65711 1048677 16772135 268400157 4294834271 68719145877 1099510700375

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 1.16.ah $\times$ 1.16.af 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 $\infty$. 1.16.ah : $$\Q(\sqrt{-15})$$. 1.16.af : $$\Q(\sqrt{-39})$$.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## 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.ak_bd_abo $2$ (not in LMFDB) 3.16.ag_ad_dk $2$ (not in LMFDB) 3.16.ae_an_fw $2$ (not in LMFDB) 3.16.e_an_afw $2$ (not in LMFDB) 3.16.g_ad_adk $2$ (not in LMFDB) 3.16.k_bd_bo $2$ (not in LMFDB) 3.16.u_gx_bjk $2$ (not in LMFDB) 3.16.ai_bj_aem $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ak_bd_abo $2$ (not in LMFDB) 3.16.ag_ad_dk $2$ (not in LMFDB) 3.16.ae_an_fw $2$ (not in LMFDB) 3.16.e_an_afw $2$ (not in LMFDB) 3.16.g_ad_adk $2$ (not in LMFDB) 3.16.k_bd_bo $2$ (not in LMFDB) 3.16.u_gx_bjk $2$ (not in LMFDB) 3.16.ai_bj_aem $3$ (not in LMFDB) 3.16.am_df_aou $4$ (not in LMFDB) 3.16.ac_n_acm $4$ (not in LMFDB) 3.16.c_n_cm $4$ (not in LMFDB) 3.16.m_df_ou $4$ (not in LMFDB) 3.16.aq_fb_azc $6$ (not in LMFDB) 3.16.ag_v_aca $6$ (not in LMFDB) 3.16.ac_f_cy $6$ (not in LMFDB) 3.16.c_f_acy $6$ (not in LMFDB) 3.16.g_v_ca $6$ (not in LMFDB) 3.16.i_bj_em $6$ (not in LMFDB) 3.16.q_fb_zc $6$ (not in LMFDB)