# 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

## 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 609 4335471 10658834928 23255678882079 51107185572367344 112477288427059953600 247157259456221726495541 542875527099808293462869007 1192557015791458154074080282096 2619990439286919822544018391652096

 $r$ 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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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)