# Properties

 Label 3.13.ar_fc_axf 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: $( 1 - 7 x + 13 x^{2} )( 1 - 10 x + 49 x^{2} - 130 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.151058869957$, $\pm0.334339837461$ 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 $A(\F_{q^r})$ 553 4169067 10652248768 23367257826171 51161662525287013 112439219264720538624 247103825522123129583457 542864486380853702327156475 1192579648654060719694371545536 2620012488495518196392536516765347

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 145 2208 28645 371117 4826116 62758497 815826469 10604911584 137859378185

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 2.13.ak_bx 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.ad_ai_df $2$ (not in LMFDB) 3.13.d_ai_adf $2$ (not in LMFDB) 3.13.r_fc_xf $2$ (not in LMFDB) 3.13.ai_bq_agg $3$ (not in LMFDB) 3.13.af_m_ap $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ad_ai_df $2$ (not in LMFDB) 3.13.d_ai_adf $2$ (not in LMFDB) 3.13.r_fc_xf $2$ (not in LMFDB) 3.13.ai_bq_agg $3$ (not in LMFDB) 3.13.af_m_ap $3$ (not in LMFDB) 3.13.ap_ei_atl $6$ (not in LMFDB) 3.13.am_de_anu $6$ (not in LMFDB) 3.13.f_m_p $6$ (not in LMFDB) 3.13.i_bq_gg $6$ (not in LMFDB) 3.13.m_de_nu $6$ (not in LMFDB) 3.13.p_ei_tl $6$ (not in LMFDB)