# Properties

 Label 2.193.aby_bmx Base Field $\F_{193}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{193}$ Dimension: $2$ L-polynomial: $( 1 - 25 x + 193 x^{2} )^{2}$ Frobenius angles: $\pm0.143734387197$, $\pm0.143734387197$ Angle rank: $1$ (numerical) Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=5x^6+133x^3+45$
• $y^2=20x^6+119x^5+92x^4+19x^3+190x^2+10x+106$
• $y^2=75x^6+177x^5+192x^4+64x^3+95x^2+92x+111$
• $y^2=127x^6+42x^5+95x^4+168x^3+x^2+26x+73$
• $y^2=126x^6+82x^5+106x^4+7x^3+45x^2+186x+94$
• $y^2=10x^6+56x^5+39x^4+30x^3+40x^2+107x+91$
• $y^2=119x^6+106x^5+50x^4+115x^3+112x^2+119x+160$
• $y^2=53x^6+144x^5+138x^4+156x^3+108x^2+x+37$
• $y^2=28x^6+92x^5+60x^4+170x^3+81x^2+56x+102$
• $y^2=89x^6+91x^5+93x^4+125x^3+24x^2+88x+87$
• $y^2=12x^6+98x^5+115x^4+48x^3+151x^2+77x+98$
• $y^2=155x^6+117x^5+83x^4+69x^3+166x^2+82x+82$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 28561 1369814121 51666021416464 1925171176753933641 71709256409245886243761 2671086347119640841444458496 99495249060631298608475922191089 3706098393377197138625638880066015625 138048458717533024006991211940954753144336 5142167038139989589404007356298021986465358121

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 144 36772 7186758 1387522756 267786496944 51682566660478 9974730725425008 1925122957865007748 371548729959925111494 71708904873488421516772

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{193}$
 The isogeny class factors as 1.193.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{193}$.

## 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 2.193.a_ajf $2$ (not in LMFDB) 2.193.by_bmx $2$ (not in LMFDB) 2.193.ax_my $3$ (not in LMFDB) 2.193.ac_ahh $3$ (not in LMFDB) 2.193.e_pa $3$ (not in LMFDB) 2.193.z_qq $3$ (not in LMFDB) 2.193.bu_bjf $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.193.a_ajf $2$ (not in LMFDB) 2.193.by_bmx $2$ (not in LMFDB) 2.193.ax_my $3$ (not in LMFDB) 2.193.ac_ahh $3$ (not in LMFDB) 2.193.e_pa $3$ (not in LMFDB) 2.193.z_qq $3$ (not in LMFDB) 2.193.bu_bjf $3$ (not in LMFDB) 2.193.a_jf $4$ (not in LMFDB) 2.193.abw_bkz $6$ (not in LMFDB) 2.193.abu_bjf $6$ (not in LMFDB) 2.193.abb_qu $6$ (not in LMFDB) 2.193.az_qq $6$ (not in LMFDB) 2.193.av_nc $6$ (not in LMFDB) 2.193.ae_pa $6$ (not in LMFDB) 2.193.a_afn $6$ (not in LMFDB) 2.193.a_os $6$ (not in LMFDB) 2.193.c_ahh $6$ (not in LMFDB) 2.193.v_nc $6$ (not in LMFDB) 2.193.x_my $6$ (not in LMFDB) 2.193.bb_qu $6$ (not in LMFDB) 2.193.bw_bkz $6$ (not in LMFDB) 2.193.a_aos $12$ (not in LMFDB) 2.193.a_fn $12$ (not in LMFDB)