# Properties

 Label 2.193.abw_bkz 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: $( 1 - 25 x + 193 x^{2} )( 1 - 23 x + 193 x^{2} )$ Frobenius angles: $\pm0.143734387197$, $\pm0.189598946136$ Angle rank: $1$ (numerical) Jacobians: 31

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 31 curves, and hence is principally polarizable:

• $y^2=187x^6+72x^5+55x^4+129x^3+55x^2+72x+187$
• $y^2=179x^6+131x^5+41x^4+73x^3+41x^2+131x+179$
• $y^2=21x^6+63x^5+186x^4+66x^3+186x^2+63x+21$
• $y^2=183x^6+79x^5+131x^4+40x^3+131x^2+79x+183$
• $y^2=131x^6+70x^5+154x^4+23x^3+154x^2+70x+131$
• $y^2=168x^6+119x^5+26x^4+101x^3+26x^2+119x+168$
• $y^2=102x^6+13x^5+189x^4+3x^3+189x^2+13x+102$
• $y^2=5x^6+168$
• $y^2=133x^6+57x^5+144x^4+82x^3+144x^2+57x+133$
• $y^2=87x^6+88x^5+179x^4+12x^3+179x^2+88x+87$
• $y^2=185x^6+32x^5+97x^4+86x^3+97x^2+32x+185$
• $y^2=57x^6+55x^5+191x^4+31x^3+191x^2+55x+57$
• $y^2=141x^6+168x^5+68x^4+142x^3+68x^2+168x+141$
• $y^2=9x^6+69x^5+24x^4+51x^3+24x^2+69x+9$
• $y^2=173x^6+77x^5+60x^4+167x^3+60x^2+77x+173$
• $y^2=21x^6+142x^5+47x^4+154x^3+47x^2+142x+21$
• $y^2=39x^6+147x^5+65x^4+130x^3+65x^2+147x+39$
• $y^2=132x^6+31x^5+35x^4+154x^3+35x^2+31x+132$
• $y^2=33x^6+77x^5+29x^4+63x^3+29x^2+77x+33$
• $y^2=77x^6+47x^5+152x^4+77x^3+152x^2+47x+77$
• and 11 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 28899 1373367177 51682553604864 1925222059351192329 71709354098056097496099 2671086347119640841444458496 99495248097890118081285007086627 3706098388334656603753467367532975625 138048458700232695871510244219046599099136 5142167038096105580768855004541634568892330377

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 146 36868 7189058 1387559428 267786861746 51682566660478 9974730628906994 1925122955245673476 371548729913362368194 71708904872876447192068

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{193}$
 The isogeny class factors as 1.193.az $\times$ 1.193.ax 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 $\overline{\F}_{193}$
 The base change of $A$ to $\F_{193^{6}}$ is 1.51682540549249.bcovba 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{193^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{193^{2}}$  The base change of $A$ to $\F_{193^{2}}$ is 1.37249.ajf $\times$ 1.37249.afn. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{193^{3}}$  The base change of $A$ to $\F_{193^{3}}$ is 1.7189057.absg $\times$ 1.7189057.bsg. 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 2.193.ac_ahh $2$ (not in LMFDB) 2.193.c_ahh $2$ (not in LMFDB) 2.193.bw_bkz $2$ (not in LMFDB) 2.193.abb_qu $3$ (not in LMFDB) 2.193.av_nc $3$ (not in LMFDB) 2.193.a_ajf $3$ (not in LMFDB) 2.193.a_afn $3$ (not in LMFDB) 2.193.a_os $3$ (not in LMFDB) 2.193.v_nc $3$ (not in LMFDB) 2.193.bb_qu $3$ (not in LMFDB) 2.193.bw_bkz $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.193.ac_ahh $2$ (not in LMFDB) 2.193.c_ahh $2$ (not in LMFDB) 2.193.bw_bkz $2$ (not in LMFDB) 2.193.abb_qu $3$ (not in LMFDB) 2.193.av_nc $3$ (not in LMFDB) 2.193.a_ajf $3$ (not in LMFDB) 2.193.a_afn $3$ (not in LMFDB) 2.193.a_os $3$ (not in LMFDB) 2.193.v_nc $3$ (not in LMFDB) 2.193.bb_qu $3$ (not in LMFDB) 2.193.bw_bkz $3$ (not in LMFDB) 2.193.aby_bmx $6$ (not in LMFDB) 2.193.abu_bjf $6$ (not in LMFDB) 2.193.az_qq $6$ (not in LMFDB) 2.193.ax_my $6$ (not in LMFDB) 2.193.ae_pa $6$ (not in LMFDB) 2.193.e_pa $6$ (not in LMFDB) 2.193.x_my $6$ (not in LMFDB) 2.193.z_qq $6$ (not in LMFDB) 2.193.bu_bjf $6$ (not in LMFDB) 2.193.by_bmx $6$ (not in LMFDB) 2.193.a_aos $12$ (not in LMFDB) 2.193.a_fn $12$ (not in LMFDB) 2.193.a_jf $12$ (not in LMFDB)