Properties

Label 3.13.ap_ee_asn
Base Field $\F_{13}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 39 x^{2} - 104 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.207527242884$, $\pm0.398160485086$
Angle rank:  $3$ (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
$A(\F_{q^r})$ 679 4577139 10810418752 23301120712779 51125792364730699 112482149567143652352 247130633442863296681471 542827922631607749748976091 1192519411382458523679552016192 2619978482014390258241850072620379

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 161 2240 28565 370859 4827956 62765303 815771525 10604375936 137857588841

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bn 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_ae_cn$2$(not in LMFDB)
3.13.b_ae_acn$2$(not in LMFDB)
3.13.p_ee_sn$2$(not in LMFDB)
3.13.ag_bk_afa$3$(not in LMFDB)
3.13.ad_m_an$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_ae_cn$2$(not in LMFDB)
3.13.b_ae_acn$2$(not in LMFDB)
3.13.p_ee_sn$2$(not in LMFDB)
3.13.ag_bk_afa$3$(not in LMFDB)
3.13.ad_m_an$3$(not in LMFDB)
3.13.an_do_apn$6$(not in LMFDB)
3.13.ak_cq_ala$6$(not in LMFDB)
3.13.d_m_n$6$(not in LMFDB)
3.13.g_bk_fa$6$(not in LMFDB)
3.13.k_cq_la$6$(not in LMFDB)
3.13.n_do_pn$6$(not in LMFDB)