# Properties

 Label 3.13.ap_ea_arl 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: $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 35 x^{2} - 104 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.126882739163$, $\pm0.439864156467$ 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 $A(\F_{q^r})$ 651 4333707 10351962432 22951801980891 51060059851663191 112561965616195243008 247191503987224050100203 542850218228437080259907259 1192544532792050915396807157312 2620011244943064284872767660218787

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 153 2144 28133 370379 4831380 62780759 815805029 10604599328 137859312753

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bj 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_ai_bl $2$ (not in LMFDB) 3.13.b_ai_abl $2$ (not in LMFDB) 3.13.p_ea_rl $2$ (not in LMFDB) 3.13.ag_bg_afi $3$ (not in LMFDB) 3.13.ad_i_abh $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ab_ai_bl $2$ (not in LMFDB) 3.13.b_ai_abl $2$ (not in LMFDB) 3.13.p_ea_rl $2$ (not in LMFDB) 3.13.ag_bg_afi $3$ (not in LMFDB) 3.13.ad_i_abh $3$ (not in LMFDB) 3.13.an_dk_aot $6$ (not in LMFDB) 3.13.ak_cm_aks $6$ (not in LMFDB) 3.13.d_i_bh $6$ (not in LMFDB) 3.13.g_bg_fi $6$ (not in LMFDB) 3.13.k_cm_ks $6$ (not in LMFDB) 3.13.n_dk_ot $6$ (not in LMFDB)