Abelian variety isogeny class 2.37.ar_fj over $\F_{37}$

• Label: 2.37.ar_fj
• Base field: $\F_{37}$
• 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:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$1 - 17 x + 139 x^{2} - 629 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.128168447427$, $\pm0.341591314401$
Angle rank:	$2$ (numerical)
Number field:	4.0.44573.1
Galois group:	$D_{4}$
Jacobians:	$9$
Isomorphism classes:	9
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$863$	$1859765$	$2580395027$	$3514779172325$	$4808376961795728$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1359$	$50943$	$1875387$	$69340966$	$2565702111$	$94932254823$	$3512485693203$	$129961782176061$	$4808584496466214$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):

• $y^2=23 x^5+31 x^4+3 x^3+19 x^2+26 x+14$
• $y^2=5 x^6+21 x^5+23 x^4+18 x^3+15 x^2+32 x+18$
• $y^2=33 x^6+3 x^5+14 x^4+24 x^3+15 x^2+25 x+33$
• $y^2=15 x^6+30 x^5+36 x^4+x^3+3 x^2+33 x+35$
• $y^2=27 x^6+20 x^5+21 x^4+27 x^3+25 x+22$
• $y^2=30 x^6+27 x^5+21 x^4+19 x^3+18 x^2+21 x+23$
• $y^2=23 x^6+21 x^5+26 x^4+18 x^3+30 x^2+26 x+2$
• $y^2=10 x^6+10 x^5+35 x^4+32 x^3+21 x^2+x+19$
• $y^2=11 x^6+15 x^5+16 x^4+x^3+30 x^2+7 x+23$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37}$.

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is 4.0.44573.1.

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
2.37.r_fj	$2$	(not in LMFDB)