Properties

Label 3.13.aq_en_aub
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 9 x + 41 x^{2} - 117 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.109149799241$, $\pm0.400911184348$
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
$A(\F_{q^r})$ 595 4210815 10399333840 23003113884975 50983602291823600 112461589614792922560 247169665734669259945615 542879775564162421878399375 1192562007348115729035492404560 2620005718030021365669969970483200

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 148 2155 28196 369823 4827073 62775214 815849444 10604754715 137859021943

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.aj_bp 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_aj_cb$2$(not in LMFDB)
3.13.c_aj_acb$2$(not in LMFDB)
3.13.q_en_ub$2$(not in LMFDB)
3.13.ah_bk_afw$3$(not in LMFDB)
3.13.ae_j_abd$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ac_aj_cb$2$(not in LMFDB)
3.13.c_aj_acb$2$(not in LMFDB)
3.13.q_en_ub$2$(not in LMFDB)
3.13.ah_bk_afw$3$(not in LMFDB)
3.13.ae_j_abd$3$(not in LMFDB)
3.13.ao_dv_aqx$6$(not in LMFDB)
3.13.al_cu_ame$6$(not in LMFDB)
3.13.e_j_bd$6$(not in LMFDB)
3.13.h_bk_fw$6$(not in LMFDB)
3.13.l_cu_me$6$(not in LMFDB)
3.13.o_dv_qx$6$(not in LMFDB)