Properties

Label 3.13.aq_el_atn
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 9 x + 39 x^{2} - 117 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0228181011636$, $\pm0.0772104791556$, $\pm0.419357734967$
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
$A(\F_{q^r})$ 581 4087335 10141926704 22738764255375 50810670439501616 112355835810658614720 247078851740814348911033 542799313739293649787795375 1192510523761141254849914881328 2619984069431691288457929961900800

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 144 2101 27868 368563 4822533 62752156 815728532 10604296903 137857882839

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.aj_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.ac_al_bn$2$(not in LMFDB)
3.13.c_al_abn$2$(not in LMFDB)
3.13.q_el_tn$2$(not in LMFDB)
3.13.ah_bi_aga$3$(not in LMFDB)
3.13.ae_h_abn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ac_al_bn$2$(not in LMFDB)
3.13.c_al_abn$2$(not in LMFDB)
3.13.q_el_tn$2$(not in LMFDB)
3.13.ah_bi_aga$3$(not in LMFDB)
3.13.ae_h_abn$3$(not in LMFDB)
3.13.ao_dt_aqn$6$(not in LMFDB)
3.13.al_cs_ama$6$(not in LMFDB)
3.13.e_h_bn$6$(not in LMFDB)
3.13.h_bi_ga$6$(not in LMFDB)
3.13.l_cs_ma$6$(not in LMFDB)
3.13.o_dt_qn$6$(not in LMFDB)