Properties

Label 3.13.aq_ep_aup
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 9 x + 43 x^{2} - 117 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.161492811255$, $\pm0.377973052280$
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
$A(\F_{q^r})$ 609 4335471 10658834928 23255678882079 51107185572367344 112477288427059953600 247157259456221726495541 542875527099808293462869007 1192557015791458154074080282096 2619990439286919822544018391652096

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 152 2209 28508 370723 4827749 62772064 815843060 10604710327 137858218007

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.aj_br 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_ah_cp$2$(not in LMFDB)
3.13.c_ah_acp$2$(not in LMFDB)
3.13.q_ep_up$2$(not in LMFDB)
3.13.ah_bm_afs$3$(not in LMFDB)
3.13.ae_l_at$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ac_ah_cp$2$(not in LMFDB)
3.13.c_ah_acp$2$(not in LMFDB)
3.13.q_ep_up$2$(not in LMFDB)
3.13.ah_bm_afs$3$(not in LMFDB)
3.13.ae_l_at$3$(not in LMFDB)
3.13.ao_dx_arh$6$(not in LMFDB)
3.13.al_cw_ami$6$(not in LMFDB)
3.13.e_l_t$6$(not in LMFDB)
3.13.h_bm_fs$6$(not in LMFDB)
3.13.l_cw_mi$6$(not in LMFDB)
3.13.o_dx_rh$6$(not in LMFDB)