Properties

Label 3.13.ap_ec_arz
Base Field $\F_{13}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 37 x^{2} - 104 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.167453355204$, $\pm0.421338968608$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

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
$A(\F_{q^r})$ 665 4454835 10580245760 23132465785275 51114869396254125 112555049423952199680 247187533981403309423465 542847876568293294044814075 1192527697090619236698821146880 2619987876467511499388924222920875

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 157 2192 28357 370779 4831084 62779751 815801509 10604449616 137858083157

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bl 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 $\F_{13}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_ag_bz$2$(not in LMFDB)
3.13.b_ag_abz$2$(not in LMFDB)
3.13.p_ec_rz$2$(not in LMFDB)
3.13.ag_bi_afe$3$(not in LMFDB)
3.13.ad_k_ax$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_ag_bz$2$(not in LMFDB)
3.13.b_ag_abz$2$(not in LMFDB)
3.13.p_ec_rz$2$(not in LMFDB)
3.13.ag_bi_afe$3$(not in LMFDB)
3.13.ad_k_ax$3$(not in LMFDB)
3.13.an_dm_apd$6$(not in LMFDB)
3.13.ak_co_akw$6$(not in LMFDB)
3.13.d_k_x$6$(not in LMFDB)
3.13.g_bi_fe$6$(not in LMFDB)
3.13.k_co_kw$6$(not in LMFDB)
3.13.n_dm_pd$6$(not in LMFDB)