# 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

## 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 581 4087335 10141926704 22738764255375 50810670439501616 112355835810658614720 247078851740814348911033 542799313739293649787795375 1192510523761141254849914881328 2619984069431691288457929961900800

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