Abelian variety isogeny class 2.89.r_jl over $\F_{89}$

• Label: 2.89.r_jl
• Base field: $\F_{89}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$1 + 17 x + 245 x^{2} + 1513 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.606732206369$, $\pm0.693807873504$
Angle rank:	$2$ (numerical)
Number field:	4.0.33535845.1
Galois group:	$D_{4}$
Jacobians:	$32$
Isomorphism classes:	32
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9697$	$64358989$	$494839781521$	$3937118996937205$	$31182630882077508352$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$107$	$8123$	$701927$	$62750691$	$5584222582$	$496979459651$	$44231337380743$	$3936588898381411$	$350356403001664403$	$31181719930004388878$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=24 x^6+85 x^5+62 x^4+42 x^3+36 x^2+11 x+36$
• $y^2=x^6+9 x^5+64 x^4+71 x^3+55 x^2+6 x+45$
• $y^2=69 x^6+67 x^4+42 x^3+47 x^2+46 x+29$
• $y^2=30 x^6+62 x^5+62 x^4+43 x^3+71 x^2+68 x+49$
• $y^2=6 x^6+11 x^5+6 x^4+42 x^3+77 x^2+84 x+69$
• $y^2=42 x^6+10 x^5+9 x^4+6 x^3+53 x^2+16 x+36$
• $y^2=60 x^6+67 x^5+50 x^4+81 x^2+11 x+2$
• $y^2=3 x^6+88 x^5+47 x^4+75 x^3+3 x^2+15 x+31$
• $y^2=49 x^6+38 x^5+37 x^4+83 x^3+48 x^2+81 x+73$
• $y^2=37 x^6+76 x^5+81 x^4+2 x^3+34 x^2+50$
• $y^2=45 x^6+26 x^5+38 x^4+80 x^3+5 x^2+6 x+53$
• $y^2=70 x^6+4 x^5+2 x^4+7 x^3+70 x^2+76 x+39$
• $y^2=20 x^6+18 x^5+73 x^4+38 x^3+11 x^2+87 x+59$
• $y^2=79 x^6+77 x^5+36 x^4+78 x^3+87 x^2+67 x+67$
• $y^2=40 x^6+45 x^5+34 x^4+83 x^3+30 x^2+29 x+77$
• $y^2=39 x^6+28 x^5+11 x^4+71 x^3+72 x^2+4 x+12$
• $y^2=37 x^6+20 x^5+61 x^4+13 x^3+83 x^2+64 x+55$
• $y^2=29 x^6+69 x^5+3 x^4+26 x^3+68 x^2+12 x+3$
• $y^2=8 x^6+80 x^5+68 x^4+42 x^3+11 x^2+77 x+50$
• $y^2=8 x^6+67 x^5+33 x^4+54 x^3+66 x^2+19 x+6$
• and

• $y^2=11 x^6+75 x^5+85 x^4+60 x^3+46 x^2+28 x+44$
• $y^2=25 x^6+72 x^5+87 x^4+56 x^3+69 x^2+16 x+71$
• $y^2=75 x^6+79 x^5+85 x^4+48 x^3+87 x^2+21 x+81$
• $y^2=27 x^6+7 x^5+49 x^4+18 x^3+8 x^2+78 x+48$
• $y^2=46 x^6+13 x^5+76 x^4+17 x^3+22 x^2+82 x+78$
• $y^2=10 x^6+53 x^5+6 x^4+64 x^3+6 x^2+67 x+38$
• $y^2=79 x^6+28 x^5+77 x^4+43 x^3+30 x^2+24 x+20$
• $y^2=29 x^6+44 x^5+14 x^4+79 x^3+51 x^2+83 x+54$
• $y^2=55 x^5+39 x^4+37 x^3+55 x^2+60 x+27$
• $y^2=18 x^6+84 x^5+50 x^4+67 x^3+49 x^2+22 x+75$
• $y^2=28 x^6+51 x^5+80 x^4+82 x^3+38 x^2+56 x+71$
• $y^2=30 x^6+84 x^5+38 x^4+24 x^3+16 x^2+58 x+10$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89}$.

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is 4.0.33535845.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.89.ar_jl	$2$	(not in LMFDB)