# Properties

 Label 3.13.ap_ec_arz 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: $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 37 x^{2} - 104 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.167453355204$, $\pm0.421338968608$ 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 $A(\F_{q^r})$ 665 4454835 10580245760 23132465785275 51114869396254125 112555049423952199680 247187533981403309423465 542847876568293294044814075 1192527697090619236698821146880 2619987876467511499388924222920875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 157 2192 28357 370779 4831084 62779751 815801509 10604449616 137858083157

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bl 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.ab_ag_bz $2$ (not in LMFDB) 3.13.b_ag_abz $2$ (not in LMFDB) 3.13.p_ec_rz $2$ (not in LMFDB) 3.13.ag_bi_afe $3$ (not in LMFDB) 3.13.ad_k_ax $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ab_ag_bz $2$ (not in LMFDB) 3.13.b_ag_abz $2$ (not in LMFDB) 3.13.p_ec_rz $2$ (not in LMFDB) 3.13.ag_bi_afe $3$ (not in LMFDB) 3.13.ad_k_ax $3$ (not in LMFDB) 3.13.an_dm_apd $6$ (not in LMFDB) 3.13.ak_co_akw $6$ (not in LMFDB) 3.13.d_k_x $6$ (not in LMFDB) 3.13.g_bi_fe $6$ (not in LMFDB) 3.13.k_co_kw $6$ (not in LMFDB) 3.13.n_dm_pd $6$ (not in LMFDB)