# Properties

 Label 3.13.ap_ef_asu 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: $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 40 x^{2} - 104 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.229660330050$, $\pm0.383259645523$ 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 $A(\F_{q^r})$ 686 4638732 10926245792 23381009108016 51114847857411366 112421676386927432448 247085646745369705463918 542818548136892226822733824 1192527624141034433696590886432 2619985668964138696373481391519212

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 163 2264 28663 370779 4825360 62753879 815757439 10604448968 137857967003

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bo 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_ad_cu $2$ (not in LMFDB) 3.13.b_ad_acu $2$ (not in LMFDB) 3.13.p_ef_su $2$ (not in LMFDB) 3.13.ag_bl_aey $3$ (not in LMFDB) 3.13.ad_n_ai $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ab_ad_cu $2$ (not in LMFDB) 3.13.b_ad_acu $2$ (not in LMFDB) 3.13.p_ef_su $2$ (not in LMFDB) 3.13.ag_bl_aey $3$ (not in LMFDB) 3.13.ad_n_ai $3$ (not in LMFDB) 3.13.an_dp_aps $6$ (not in LMFDB) 3.13.ak_cr_alc $6$ (not in LMFDB) 3.13.d_n_i $6$ (not in LMFDB) 3.13.g_bl_ey $6$ (not in LMFDB) 3.13.k_cr_lc $6$ (not in LMFDB) 3.13.n_dp_ps $6$ (not in LMFDB)