# Properties

 Label 2.163.aby_bkp Base Field $\F_{163}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{163}$ Dimension: $2$ L-polynomial: $( 1 - 25 x + 163 x^{2} )^{2}$ Frobenius angles: $\pm0.0652307277549$, $\pm0.0652307277549$ Angle rank: $1$ (numerical) Jacobians: 2

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

• $y^2=2x^6+59x^3+122$
• $y^2=129x^6+4x^5+27x^4+153x^3+27x^2+4x+129$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 19321 690165441 18725940713104 498260220088499001 13239554876288343679561 351763779282648898653686016 9346014648606910149640314343249 248314265736217549195510067913275625 6597461724572951880052841378186437791376 175287960541966610007672298774685592764750721

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 114 25972 4323948 705839236 115062912294 18755363780998 3057125211164418 498311414511757828 81224760543592981524 13239635967230078250772

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The isogeny class factors as 1.163.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{163}$.

## 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 2.163.a_aln $2$ (not in LMFDB) 2.163.by_bkp $2$ (not in LMFDB) 2.163.ar_ew $3$ (not in LMFDB) 2.163.ai_adv $3$ (not in LMFDB) 2.163.q_pa $3$ (not in LMFDB) 2.163.z_ru $3$ (not in LMFDB) 2.163.bi_xr $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.163.a_aln $2$ (not in LMFDB) 2.163.by_bkp $2$ (not in LMFDB) 2.163.ar_ew $3$ (not in LMFDB) 2.163.ai_adv $3$ (not in LMFDB) 2.163.q_pa $3$ (not in LMFDB) 2.163.z_ru $3$ (not in LMFDB) 2.163.bi_xr $3$ (not in LMFDB) 2.163.a_ln $4$ (not in LMFDB) 2.163.abq_bcx $6$ (not in LMFDB) 2.163.abi_xr $6$ (not in LMFDB) 2.163.abh_ug $6$ (not in LMFDB) 2.163.az_ru $6$ (not in LMFDB) 2.163.aq_pa $6$ (not in LMFDB) 2.163.aj_hi $6$ (not in LMFDB) 2.163.a_bl $6$ (not in LMFDB) 2.163.a_kc $6$ (not in LMFDB) 2.163.i_adv $6$ (not in LMFDB) 2.163.j_hi $6$ (not in LMFDB) 2.163.r_ew $6$ (not in LMFDB) 2.163.bh_ug $6$ (not in LMFDB) 2.163.bq_bcx $6$ (not in LMFDB) 2.163.a_akc $12$ (not in LMFDB) 2.163.a_abl $12$ (not in LMFDB)