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

• Label: 2.89.abf_px
• 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:	$1 - 31 x + 413 x^{2} - 2759 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.108051792485$, $\pm0.253156413300$
Angle rank:	$2$ (numerical)
Number field:	4.0.3159765.2
Galois group:	$D_{4}$
Jacobians:	$10$
Isomorphism classes:	10
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5545$	$61688125$	$497222018665$	$3937371964103125$	$31182385361885689600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$59$	$7787$	$705311$	$62754723$	$5584178614$	$496981840547$	$44231334451951$	$3936588794884003$	$350356404127729859$	$31181719941884537102$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=5 x^6+53 x^5+26 x^4+24 x^3+49 x^2+43 x+2$
• $y^2=51 x^6+28 x^5+30 x^4+24 x^3+15 x^2+83 x+69$
• $y^2=12 x^6+10 x^5+16 x^4+16 x^3+51 x^2+27 x+82$
• $y^2=20 x^6+39 x^5+76 x^4+32 x^3+17 x^2+84 x+61$
• $y^2=30 x^6+66 x^5+67 x^4+19 x^3+23 x^2+66 x+62$
• $y^2=61 x^6+71 x^5+86 x^4+51 x^3+39 x^2+29 x+24$
• $y^2=44 x^6+67 x^4+86 x^3+79 x^2+4 x+39$
• $y^2=71 x^6+18 x^5+85 x^4+43 x^3+67 x^2+72 x+30$
• $y^2=75 x^6+51 x^5+6 x^4+9 x^3+53 x^2+9 x+58$
• $y^2=31 x^6+26 x^5+75 x^4+31 x^3+77 x^2+16 x+57$

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 4.0.3159765.2.

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.bf_px	$2$	(not in LMFDB)