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

• Label: 2.37.j_de
• 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:	$( 1 + x + 37 x^{2} )( 1 + 8 x + 37 x^{2} )$
	$1 + 9 x + 82 x^{2} + 333 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.526194466411$, $\pm0.728426571754$
Angle rank:	$2$ (numerical)
Jacobians:	$28$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1794$	$1991340$	$2541251232$	$3512564452800$	$4808547641558154$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$47$	$1453$	$50168$	$1874209$	$69343427$	$2565775546$	$94932128135$	$3512473278241$	$129961759158776$	$4808584553699893$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

• $y^2=21 x^6+33 x^5+29 x^4+18 x^3+10 x^2+3 x+16$
• $y^2=33 x^6+10 x^5+30 x^4+x^3+3 x^2+15 x+22$
• $y^2=31 x^6+6 x^5+8 x^4+27 x^3+5 x^2+3 x+30$
• $y^2=28 x^6+31 x^5+19 x^4+3 x^3+x^2+26 x+2$
• $y^2=3 x^6+36 x^5+9 x^4+10 x^3+4 x^2+18$
• $y^2=9 x^6+29 x^5+10 x^4+24 x^2+36 x+12$
• $y^2=34 x^6+35 x^5+14 x^4+8 x^3+33 x^2+22 x+5$
• $y^2=19 x^6+30 x^5+7 x^4+26 x^3+x^2+6 x+21$
• $y^2=10 x^6+17 x^5+19 x^4+9 x^3+15 x^2+16 x+10$
• $y^2=21 x^6+15 x^5+8 x^4+7 x^3+25 x^2+19 x+12$
• $y^2=x^6+2 x^5+14 x^4+12 x^3+13 x^2+8 x+24$
• $y^2=13 x^6+30 x^5+30 x^4+6 x^3+30 x^2+8 x+10$
• $y^2=30 x^6+18 x^5+22 x^4+17 x^3+15 x^2+32 x+36$
• $y^2=27 x^6+15 x^5+21 x^4+28 x^3+28 x^2+19 x+19$
• $y^2=9 x^6+19 x^5+3 x^4+23 x^3+18 x^2+28 x+21$
• $y^2=10 x^6+36 x^5+25 x^4+15 x^3+34 x^2+24 x+4$
• $y^2=5 x^6+30 x^5+13 x^4+2 x^3+9 x^2+8 x+23$
• $y^2=19 x^6+15 x^5+23 x^4+32 x^3+28 x^2+21 x+28$
• $y^2=16 x^6+24 x^5+3 x^4+5 x^3+11 x^2+17 x+25$
• $y^2=26 x^6+33 x^5+25 x^4+20 x^3+11 x^2+35 x+33$
• $y^2=28 x^5+30 x^4+23 x^3+23 x^2+7 x+26$
• $y^2=16 x^6+22 x^5+17 x^4+17 x^3+36 x^2+14 x+33$
• $y^2=7 x^6+18 x^5+31 x^4+21 x^3+24 x^2+14 x+18$
• $y^2=14 x^6+15 x^5+36 x^4+2 x^3+29 x^2+4 x+19$
• $y^2=24 x^6+8 x^5+28 x^4+7 x^3+10 x^2+20 x+4$
• $y^2=32 x^6+21 x^5+4 x^4+12 x^3+27 x^2+20 x+30$
• $y^2=36 x^6+17 x^4+15 x^3+11 x^2+12 x+1$
• $y^2=9 x^6+7 x^5+20 x^4+16 x^3+25 x^2+4 x+34$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.b $\times$ 1.37.i 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.b : \(\Q(\sqrt{-3}) \).
• 1.37.i : \(\Q(\sqrt{-21}) \).

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.aj_de	$2$	(not in LMFDB)
2.37.ah_co	$2$	(not in LMFDB)
2.37.h_co	$2$	(not in LMFDB)
2.37.ad_ao	$3$	(not in LMFDB)
2.37.s_fy	$3$	(not in LMFDB)