Properties

Label 3.16.at_gf_abfc
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:  $( 1 - 4 x )^{2}( 1 - 11 x + 57 x^{2} - 176 x^{3} + 256 x^{4} )$
Frobenius angles:  $0$, $0$, $\pm0.0728689886706$, $\pm0.368631800070$
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
$A(\F_{q^r})$ 1143 14316075 66931501392 278133063001875 1147765479839878383 4717796890056573960000 19341117087390655008627303 79228754581661894026424851875 324518583758652220931538185228112 1329225728567999629128944922737476875

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 218 3991 64754 1043878 16760975 268411918 4294999394 68719483111 1099509752378

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 2.16.al_cf 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.ad_ap_ea$2$(not in LMFDB)
3.16.d_ap_aea$2$(not in LMFDB)
3.16.t_gf_bfc$2$(not in LMFDB)
3.16.ah_bd_aeu$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ad_ap_ea$2$(not in LMFDB)
3.16.d_ap_aea$2$(not in LMFDB)
3.16.t_gf_bfc$2$(not in LMFDB)
3.16.ah_bd_aeu$3$(not in LMFDB)
3.16.al_cv_ano$4$(not in LMFDB)
3.16.l_cv_no$4$(not in LMFDB)
3.16.ap_en_awi$6$(not in LMFDB)
3.16.h_bd_eu$6$(not in LMFDB)
3.16.p_en_wi$6$(not in LMFDB)