Properties

Label 3.16.au_gv_abiu
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:  $( 1 - 4 x )^{2}( 1 - 12 x + 65 x^{2} - 192 x^{3} + 256 x^{4} )$
Frobenius angles:  $0$, $0$, $\pm0.0826163580681$, $\pm0.320878822416$
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
$A(\F_{q^r})$ 1062 13965300 67160711142 279547543828800 1149343372460251782 4717788853900646733300 19339337645703160662929238 79227421077482840321872876800 324520116587361235231008975262998 1329229636192222503973373548304032500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 211 4005 65087 1045317 16760947 268387221 4294927103 68719807701 1099512984691

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 2.16.am_cn 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.ae_ap_fg$2$(not in LMFDB)
3.16.e_ap_afg$2$(not in LMFDB)
3.16.u_gv_biu$2$(not in LMFDB)
3.16.ai_bh_aeu$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ae_ap_fg$2$(not in LMFDB)
3.16.e_ap_afg$2$(not in LMFDB)
3.16.u_gv_biu$2$(not in LMFDB)
3.16.ai_bh_aeu$3$(not in LMFDB)
3.16.am_dd_aou$4$(not in LMFDB)
3.16.m_dd_ou$4$(not in LMFDB)
3.16.aq_ez_ayu$6$(not in LMFDB)
3.16.i_bh_eu$6$(not in LMFDB)
3.16.q_ez_yu$6$(not in LMFDB)