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

• Label: 2.37.ar_fo
• Base field: $\F_{37}$
• 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:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 37 x^{2} )( 1 - 7 x + 37 x^{2} )$
	$1 - 17 x + 144 x^{2} - 629 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.192861133077$, $\pm0.304847772502$
Angle rank:	$2$ (numerical)
Jacobians:	$10$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$868$	$1874880$	$2593431232$	$3520312185600$	$4809644273847028$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1369$	$51198$	$1878337$	$69359241$	$2565728566$	$94931594373$	$3512478233953$	$129961740206886$	$4808584374285889$

Jacobians and polarizations

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

• $y^2=32 x^6+19 x^5+14 x^3+26 x+14$
• $y^2=31 x^6+36 x^5+24 x^4+14 x^3+29 x^2+12 x+2$
• $y^2=4 x^6+12 x^5+12 x^4+7 x^3+29 x+30$
• $y^2=23 x^6+19 x^5+30 x^4+36 x^3+31 x^2+30 x+22$
• $y^2=31 x^6+14 x^5+18 x^4+35 x^3+17 x^2+24 x+29$
• $y^2=23 x^6+5 x^5+21 x^4+22 x^3+16 x^2+17 x+28$
• $y^2=14 x^6+17 x^5+6 x^4+19 x^3+14 x^2+31 x+13$
• $y^2=35 x^6+21 x^5+23 x^4+7 x^3+17 x^2+15 x+20$
• $y^2=31 x^6+4 x^5+15 x^4+28 x^3+18 x^2+x+6$
• $y^2=26 x^6+3 x^5+24 x^4+8 x^3+32 x^2+27 x+13$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ak $\times$ 1.37.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.37.ak : \(\Q(\sqrt{-3}) \).
• 1.37.ah : \(\Q(\sqrt{-11}) \).

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
2.37.ad_e	$2$	(not in LMFDB)
2.37.d_e	$2$	(not in LMFDB)
2.37.r_fo	$2$	(not in LMFDB)
2.37.ai_dd	$3$	(not in LMFDB)
2.37.e_ad	$3$	(not in LMFDB)