Abelian variety isogeny class 2.109.abk_uw over $\F_{109}$

• Label: 2.109.abk_uw
• Base field: $\F_{109}$
• 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:	$\F_{109}$
Dimension:	$2$
L-polynomial:	$( 1 - 18 x + 109 x^{2} )^{2}$
	$1 - 36 x + 542 x^{2} - 3924 x^{3} + 11881 x^{4}$
Frobenius angles:	$\pm0.169184306747$, $\pm0.169184306747$
Angle rank:	$1$ (numerical)
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8464$	$138674176$	$1677242567056$	$19929163150393344$	$236743124620706193424$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$11670$	$1295138$	$141183214$	$15386678714$	$1677105285126$	$182803957349906$	$19925626667733214$	$2171893279023034922$	$236736367424324172150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=94 x^6+11 x^5+44 x^4+28 x^3+44 x^2+11 x+94$
• $y^2=52 x^6+6 x^4+6 x^2+52$
• $y^2=13 x^6+18 x^5+x^4+96 x^3+x^2+18 x+13$
• $y^2=92 x^6+26 x^5+10 x^4+9 x^3+103 x^2+5 x+90$
• $y^2=23 x^6+26 x^5+82 x^4+52 x^3+82 x^2+26 x+23$
• $y^2=101 x^6+35 x^5+76 x^4+58 x^3+76 x^2+35 x+101$
• $y^2=14 x^6+105 x^5+46 x^4+100 x^3+46 x^2+105 x+14$
• $y^2=4 x^6+97 x^4+97 x^2+4$
• $y^2=48 x^6+54 x^5+28 x^4+34 x^3+76 x^2+43 x+30$
• $y^2=67 x^6+73 x^5+27 x^4+45 x^3+5 x^2+97 x+62$
• $y^2=107 x^6+69 x^5+17 x^4+68 x^3+101 x^2+52 x+12$
• $y^2=11 x^6+62 x^4+62 x^2+11$
• $y^2=10 x^6+20 x^5+32 x^4+67 x^3+32 x^2+20 x+10$
• $y^2=7 x^6+82 x^4+82 x^2+7$
• $y^2=94 x^6+85 x^5+79 x^4+75 x^3+79 x^2+85 x+94$
• $y^2=54 x^6+85 x^5+2 x^4+103 x^3+91 x^2+104 x+69$
• $y^2=16 x^6+23 x^5+48 x^4+60 x^3+96 x^2+27 x+89$
• $y^2=47 x^6+102 x^5+17 x^4+94 x^3+17 x^2+102 x+47$
• $y^2=92 x^6+40 x^5+26 x^4+9 x^3+28 x^2+73 x+96$
• $y^2=47 x^6+51 x^5+71 x^4+36 x^3+11 x^2+4 x+56$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{109}$.

Endomorphism algebra over $\F_{109}$

The isogeny class factors as 1.109.as^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
2.109.a_aec	$2$	(not in LMFDB)
2.109.bk_uw	$2$	(not in LMFDB)
2.109.s_ih	$3$	(not in LMFDB)