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

• Label: 2.37.o_dz
• 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:	$1 + 14 x + 103 x^{2} + 518 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.566626788045$, $\pm0.892012654627$
Angle rank:	$2$ (numerical)
Number field:	4.0.58025.1
Galois group:	$D_{4}$
Jacobians:	$32$
Isomorphism classes:	32

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2005$	$1886705$	$2564346880$	$3507960040025$	$4809439769435125$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$1380$	$50626$	$1871748$	$69356292$	$2565803190$	$94930820596$	$3512483252868$	$129961738775962$	$4808584437568900$

Jacobians and polarizations

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

• $y^2=11 x^6+36 x^5+x^4+x^3+35 x^2+3$
• $y^2=29 x^6+11 x^5+7 x^4+19 x^3+17 x^2+6 x+25$
• $y^2=3 x^6+32 x^5+20 x^4+34 x^3+20 x^2+24 x+9$
• $y^2=2 x^6+9 x^5+17 x^4+16 x^3+19 x+27$
• $y^2=6 x^6+10 x^5+4 x^4+31 x^3+21 x^2+8 x+28$
• $y^2=9 x^6+5 x^5+20 x^4+19 x^3+30 x^2+29 x+19$
• $y^2=26 x^6+33 x^5+23 x^4+3 x^3+34 x^2+10 x+7$
• $y^2=25 x^6+17 x^5+19 x^4+28 x^3+14 x^2+32 x+12$
• $y^2=18 x^6+23 x^5+8 x^3+28 x^2+12 x+31$
• $y^2=21 x^6+28 x^5+35 x^4+13 x^3+36 x^2+x+3$
• $y^2=36 x^6+20 x^5+4 x^4+25 x^3+9 x^2+7$
• $y^2=19 x^6+26 x^5+11 x^4+x^3+9 x^2+20 x+32$
• $y^2=8 x^6+13 x^5+x^4+19 x^3+29 x^2+5 x+9$
• $y^2=5 x^6+23 x^4+17 x^3+31 x^2+7 x+32$
• $y^2=26 x^6+24 x^5+30 x^4+32 x^3+9 x^2+10 x+27$
• $y^2=19 x^6+12 x^5+13 x^4+5 x^3+9 x^2+36$
• $y^2=3 x^6+5 x^5+4 x^4+14 x^3+31 x^2+34 x+16$
• $y^2=18 x^6+19 x^5+21 x^4+29 x^3+16 x^2+22 x+9$
• $y^2=8 x^6+3 x^5+23 x^4+10 x^3+16 x^2+14 x+7$
• $y^2=x^6+30 x^5+10 x^4+33 x^3+32 x^2+34 x+27$
• and

• $y^2=4 x^6+33 x^5+35 x^4+3 x^3+30 x^2+x+7$
• $y^2=2 x^6+14 x^5+22 x^4+8 x^3+27 x^2+10 x+2$
• $y^2=2 x^6+30 x^5+13 x^4+10 x^3+14 x^2+24 x+7$
• $y^2=24 x^6+9 x^5+34 x^4+15 x^3+11 x^2+8 x+34$
• $y^2=28 x^6+9 x^5+29 x^4+17 x^3+20 x^2+18 x+10$
• $y^2=32 x^6+15 x^5+12 x^4+19 x^3+9 x^2+8 x+4$
• $y^2=26 x^6+33 x^5+15 x^4+29 x^3+16 x^2+9 x+11$
• $y^2=19 x^6+19 x^5+14 x^4+31 x^3+32 x^2+22 x+2$
• $y^2=33 x^6+35 x^5+26 x^4+25 x^3+29 x^2+33 x+16$
• $y^2=21 x^6+18 x^5+2 x^4+28 x^3+14 x^2+30 x+9$
• $y^2=21 x^6+22 x^5+23 x^4+23 x^3+16 x^2+20 x+1$
• $y^2=29 x^6+26 x^5+9 x^4+29 x^3+21 x^2+35 x+3$

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 4.0.58025.1.

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.ao_dz	$2$	(not in LMFDB)