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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1080 14256000 68374280520 282218904000000 1153030973368527000 4720935756600279216000 19340269453314591891183720 79225708645026070037856000000 324516991224084672707130388868280 1329226874624178336332923025310000000

Point counts of the (virtual) curve

$r$ 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:
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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)