Properties

Label 3.13.ap_eb_ars
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 36 x^{2} - 104 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.147614849952$, $\pm0.431019279425$
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
$A(\F_{q^r})$ 658 4394124 10465874272 23043647504304 51092959908644458 112566898734816363264 247196797297426767647794 542852706140314393827225600 1192536973193493054737051483104 2619999422337770858415665380625964

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 155 2168 28247 370619 4831592 62782103 815808767 10604532104 137858690675

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bk 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_ah_bs$2$(not in LMFDB)
3.13.b_ah_abs$2$(not in LMFDB)
3.13.p_eb_rs$2$(not in LMFDB)
3.13.ag_bh_afg$3$(not in LMFDB)
3.13.ad_j_abc$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_ah_bs$2$(not in LMFDB)
3.13.b_ah_abs$2$(not in LMFDB)
3.13.p_eb_rs$2$(not in LMFDB)
3.13.ag_bh_afg$3$(not in LMFDB)
3.13.ad_j_abc$3$(not in LMFDB)
3.13.an_dl_aoy$6$(not in LMFDB)
3.13.ak_cn_aku$6$(not in LMFDB)
3.13.d_j_bc$6$(not in LMFDB)
3.13.g_bh_fg$6$(not in LMFDB)
3.13.k_cn_ku$6$(not in LMFDB)
3.13.n_dl_oy$6$(not in LMFDB)