# Properties

 Label 3.13.ap_eb_ars 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: $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 36 x^{2} - 104 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.147614849952$, $\pm0.431019279425$ 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 $A(\F_{q^r})$ 658 4394124 10465874272 23043647504304 51092959908644458 112566898734816363264 247196797297426767647794 542852706140314393827225600 1192536973193493054737051483104 2619999422337770858415665380625964

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 155 2168 28247 370619 4831592 62782103 815808767 10604532104 137858690675

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bk 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_ah_bs $2$ (not in LMFDB) 3.13.b_ah_abs $2$ (not in LMFDB) 3.13.p_eb_rs $2$ (not in LMFDB) 3.13.ag_bh_afg $3$ (not in LMFDB) 3.13.ad_j_abc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ab_ah_bs $2$ (not in LMFDB) 3.13.b_ah_abs $2$ (not in LMFDB) 3.13.p_eb_rs $2$ (not in LMFDB) 3.13.ag_bh_afg $3$ (not in LMFDB) 3.13.ad_j_abc $3$ (not in LMFDB) 3.13.an_dl_aoy $6$ (not in LMFDB) 3.13.ak_cn_aku $6$ (not in LMFDB) 3.13.d_j_bc $6$ (not in LMFDB) 3.13.g_bh_fg $6$ (not in LMFDB) 3.13.k_cn_ku $6$ (not in LMFDB) 3.13.n_dl_oy $6$ (not in LMFDB)