# Properties

 Label 3.13.ap_dx_aqq 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: $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 32 x^{2} - 104 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.0370621216586$, $\pm0.0772104791556$, $\pm0.462937878341$ Angle rank: $2$ (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 $A(\F_{q^r})$ 630 4154220 10012857120 22657863639600 50895254892840750 112443633371876002560 247075057715527390608630 542769149576876086635724800 1192514971681397866466643104160 2620006291359067191715284001015500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 147 2072 27767 369179 4826304 62751191 815683199 10604336456 137859052107

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bg and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{4}}$ is 1.28561.alq 2 $\times$ 1.28561.ahj. The endomorphism algebra for each factor is: 1.28561.alq 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-10})$$$)$ 1.28561.ahj : $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{2}}$  The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 2.169.a_alq. The endomorphism algebra for each factor is:

## 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_al_q $2$ (not in LMFDB) 3.13.b_al_aq $2$ (not in LMFDB) 3.13.p_dx_qq $2$ (not in LMFDB) 3.13.ag_bd_afo $3$ (not in LMFDB) 3.13.ad_f_abw $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ab_al_q $2$ (not in LMFDB) 3.13.b_al_aq $2$ (not in LMFDB) 3.13.p_dx_qq $2$ (not in LMFDB) 3.13.ag_bd_afo $3$ (not in LMFDB) 3.13.ad_f_abw $3$ (not in LMFDB) 3.13.an_dh_aoe $6$ (not in LMFDB) 3.13.ak_cj_akm $6$ (not in LMFDB) 3.13.d_f_bw $6$ (not in LMFDB) 3.13.g_bd_fo $6$ (not in LMFDB) 3.13.k_cj_km $6$ (not in LMFDB) 3.13.n_dh_oe $6$ (not in LMFDB) 3.13.ah_h_bq $8$ (not in LMFDB) 3.13.ah_t_abq $8$ (not in LMFDB) 3.13.h_h_abq $8$ (not in LMFDB) 3.13.h_t_bq $8$ (not in LMFDB) 3.13.af_h_be $24$ (not in LMFDB) 3.13.af_t_abe $24$ (not in LMFDB) 3.13.ac_h_m $24$ (not in LMFDB) 3.13.ac_t_am $24$ (not in LMFDB) 3.13.c_h_am $24$ (not in LMFDB) 3.13.c_t_m $24$ (not in LMFDB) 3.13.f_h_abe $24$ (not in LMFDB) 3.13.f_t_be $24$ (not in LMFDB)