# Properties

 Label 3.16.at_gi_abga Base Field $\F_{2^{4}}$ Dimension $3$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 7 x + 16 x^{2} )( 1 - 4 x + 16 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.160861246510$, $\pm0.333333333333$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 1170 14742000 68585312250 281333762892000 1151383396862234250 4720046579850542850000 19341449817608086929339930 79228454097498072812571672000 324518958233008035416524588142250 1329225879739998750287299160388750000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 224 4090 65504 1047178 16768976 268416538 4294983104 68719562410 1099509877424

## 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.ae 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.ae : $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{2^{4}}$
 The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc 2 $\times$ 1.16777216.mbf. The endomorphism algebra for each factor is: 1.16777216.amdc 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16777216.mbf : $$\Q(\sqrt{-15})$$.
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 $\times$ 1.256.ar $\times$ 1.256.q. The endomorphism algebra for each factor is: 1.256.abg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.256.ar : $$\Q(\sqrt{-15})$$. 1.256.q : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.ah $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.aey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4096.ah : $$\Q(\sqrt{-15})$$. 1.4096.ey : 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 3.16.al_bs_aey $2$ (not in LMFDB) 3.16.af_ae_cm $2$ (not in LMFDB) 3.16.ad_am_ey $2$ (not in LMFDB) 3.16.d_am_aey $2$ (not in LMFDB) 3.16.f_ae_acm $2$ (not in LMFDB) 3.16.l_bs_ey $2$ (not in LMFDB) 3.16.t_gi_bga $2$ (not in LMFDB) 3.16.ah_aq_iq $3$ (not in LMFDB) 3.16.ah_bg_aei $3$ (not in LMFDB) 3.16.f_ae_acm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.al_bs_aey $2$ (not in LMFDB) 3.16.af_ae_cm $2$ (not in LMFDB) 3.16.ad_am_ey $2$ (not in LMFDB) 3.16.d_am_aey $2$ (not in LMFDB) 3.16.f_ae_acm $2$ (not in LMFDB) 3.16.l_bs_ey $2$ (not in LMFDB) 3.16.t_gi_bga $2$ (not in LMFDB) 3.16.ah_aq_iq $3$ (not in LMFDB) 3.16.ah_bg_aei $3$ (not in LMFDB) 3.16.f_ae_acm $3$ (not in LMFDB) 3.16.al_cy_ano $4$ (not in LMFDB) 3.16.ad_u_ads $4$ (not in LMFDB) 3.16.d_u_ds $4$ (not in LMFDB) 3.16.l_cy_no $4$ (not in LMFDB) 3.16.ax_iq_abto $6$ (not in LMFDB) 3.16.ap_eq_awu $6$ (not in LMFDB) 3.16.aj_a_ge $6$ (not in LMFDB) 3.16.ab_i_dc $6$ (not in LMFDB) 3.16.b_i_adc $6$ (not in LMFDB) 3.16.h_aq_aiq $6$ (not in LMFDB) 3.16.h_bg_ei $6$ (not in LMFDB) 3.16.j_a_age $6$ (not in LMFDB) 3.16.p_eq_wu $6$ (not in LMFDB) 3.16.x_iq_bto $6$ (not in LMFDB) 3.16.ap_ea_asm $12$ (not in LMFDB) 3.16.ah_a_ei $12$ (not in LMFDB) 3.16.ah_bw_aiq $12$ (not in LMFDB) 3.16.ab_ai_abg $12$ (not in LMFDB) 3.16.b_ai_bg $12$ (not in LMFDB) 3.16.h_a_aei $12$ (not in LMFDB) 3.16.h_bw_iq $12$ (not in LMFDB) 3.16.p_ea_sm $12$ (not in LMFDB) 3.16.ah_q_a $24$ (not in LMFDB) 3.16.h_q_a $24$ (not in LMFDB) 3.16.al_ci_ajg $30$ (not in LMFDB) 3.16.ad_e_q $30$ (not in LMFDB) 3.16.d_e_aq $30$ (not in LMFDB) 3.16.l_ci_jg $30$ (not in LMFDB)