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

• Label: 2.37.ar_fi
• 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:	$1 - 17 x + 138 x^{2} - 629 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.115582363187$, $\pm0.346918190411$
Angle rank:	$2$ (numerical)
Number field:	4.0.2365308.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$862$	$1856748$	$2577790312$	$3513650499264$	$4808088171224662$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1357$	$50892$	$1874785$	$69336801$	$2565686266$	$94932248397$	$3512485950625$	$129961783552092$	$4808584508223637$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=6 x^6+26 x^5+6 x^4+5 x^3+7 x^2+25 x+5$
• $y^2=11 x^6+30 x^5+26 x^4+11 x^3+23 x^2+5 x+29$
• $y^2=31 x^6+5 x^5+15 x^4+12 x^3+23 x^2+28 x+26$
• $y^2=7 x^6+21 x^5+x^4+31 x^3+28 x^2+9 x+32$
• $y^2=13 x^6+33 x^5+14 x^4+25 x^3+10 x^2+2 x+10$
• $y^2=2 x^6+7 x^5+28 x^4+27 x^3+13 x^2+18 x+18$
• $y^2=9 x^6+12 x^5+34 x^4+36 x^3+34 x^2+13 x+36$
• $y^2=16 x^6+30 x^5+32 x^4+2 x^3+18 x^2+15 x+32$
• $y^2=8 x^6+36 x^5+19 x^4+15 x^3+5 x^2+6 x+12$
• $y^2=29 x^6+5 x^5+21 x^4+10 x^3+34 x^2+7 x+25$
• $y^2=20 x^6+27 x^5+34 x^4+4 x^3+16 x^2+26 x+5$
• $y^2=35 x^6+33 x^5+15 x^4+10 x^3+18 x^2+6 x+31$

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 4.0.2365308.1.

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_fi	$2$	(not in LMFDB)