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

• Label: 2.37.ab_cq
• 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:	$( 1 - 3 x + 37 x^{2} )( 1 + 2 x + 37 x^{2} )$
	$1 - x + 68 x^{2} - 37 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.420687118444$, $\pm0.552568456711$
Angle rank:	$2$ (numerical)
Jacobians:	$45$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1400$	$2066400$	$2570422400$	$3505647600000$	$4808338770935000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$1505$	$50746$	$1870513$	$69340417$	$2565789590$	$94931920141$	$3512480065153$	$129961744667122$	$4808584250824025$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 45 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ad $\times$ 1.37.c 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.ad : \(\Q(\sqrt{-139}) \).
• 1.37.c : \(\Q(\sqrt{-1}) \).

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.af_dc	$2$	(not in LMFDB)
2.37.b_cq	$2$	(not in LMFDB)
2.37.f_dc	$2$	(not in LMFDB)