# Properties

 Label 3.13.ap_ee_asn 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: $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 39 x^{2} - 104 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.207527242884$, $\pm0.398160485086$ 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 $A(\F_{q^r})$ 679 4577139 10810418752 23301120712779 51125792364730699 112482149567143652352 247130633442863296681471 542827922631607749748976091 1192519411382458523679552016192 2619978482014390258241850072620379

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 161 2240 28565 370859 4827956 62765303 815771525 10604375936 137857588841

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 2.13.ai_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.ab_ae_cn $2$ (not in LMFDB) 3.13.b_ae_acn $2$ (not in LMFDB) 3.13.p_ee_sn $2$ (not in LMFDB) 3.13.ag_bk_afa $3$ (not in LMFDB) 3.13.ad_m_an $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ab_ae_cn $2$ (not in LMFDB) 3.13.b_ae_acn $2$ (not in LMFDB) 3.13.p_ee_sn $2$ (not in LMFDB) 3.13.ag_bk_afa $3$ (not in LMFDB) 3.13.ad_m_an $3$ (not in LMFDB) 3.13.an_do_apn $6$ (not in LMFDB) 3.13.ak_cq_ala $6$ (not in LMFDB) 3.13.d_m_n $6$ (not in LMFDB) 3.13.g_bk_fa $6$ (not in LMFDB) 3.13.k_cq_la $6$ (not in LMFDB) 3.13.n_do_pn $6$ (not in LMFDB)