Abelian variety isogeny class 2.193.abw_bkz over $\F_{193}$

• Label: 2.193.abw_bkz
• Base field: $\F_{193}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{193}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x + 193 x^{2} )( 1 - 23 x + 193 x^{2} )$
	$1 - 48 x + 961 x^{2} - 9264 x^{3} + 37249 x^{4}$
Frobenius angles:	$\pm0.143734387197$, $\pm0.189598946136$
Angle rank:	$1$ (numerical)
Jacobians:	$31$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$28899$	$1373367177$	$51682553604864$	$1925222059351192329$	$71709354098056097496099$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 31 curves (of which all are hyperelliptic):

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

• $y^2=11 x^6+95 x^5+67 x^4+175 x^3+67 x^2+95 x+11$
• $y^2=148 x^6+139 x^5+29 x^4+70 x^3+29 x^2+139 x+148$
• $y^2=167 x^6+45 x^5+54 x^4+73 x^3+54 x^2+45 x+167$
• $y^2=76 x^6+131 x^5+153 x^4+110 x^3+153 x^2+131 x+76$
• $y^2=28 x^6+62 x^5+177 x^4+135 x^3+177 x^2+62 x+28$
• $y^2=54 x^6+90 x^5+11 x^4+26 x^3+11 x^2+90 x+54$
• $y^2=38 x^6+121 x^5+71 x^4+114 x^3+71 x^2+121 x+38$
• $y^2=141 x^6+27 x^5+41 x^4+8 x^3+41 x^2+27 x+141$
• $y^2=118 x^6+152 x^5+184 x^4+190 x^3+184 x^2+152 x+118$
• $y^2=166 x^6+168 x^5+131 x^4+36 x^3+131 x^2+168 x+166$
• $y^2=114 x^6+104 x^5+172 x^4+37 x^3+172 x^2+104 x+114$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{193^{6}}$.

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:

• 1.193.az : \(\Q(\sqrt{-3}) \).
• 1.193.ax : \(\Q(\sqrt{-3}) \).

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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:

  • 1.37249.ajf : \(\Q(\sqrt{-3}) \).
  • 1.37249.afn : \(\Q(\sqrt{-3}) \).
• 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:

  • 1.7189057.absg : \(\Q(\sqrt{-3}) \).
  • 1.7189057.bsg : \(\Q(\sqrt{-3}) \).

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)