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

• Label: 2.37.ai_dm
• 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:	$( 1 - 4 x + 37 x^{2} )^{2}$
	$1 - 8 x + 90 x^{2} - 296 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.393356479550$, $\pm0.393356479550$
Angle rank:	$1$ (numerical)
Jacobians:	$26$
Cyclic group of points:	no
Non-cyclic primes:	$2, 17$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1156$	$2039184$	$2604469156$	$3510137143296$	$4806287558831236$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$1486$	$51414$	$1872910$	$69310830$	$2565640222$	$94932758118$	$3512486166814$	$129961734050238$	$4808584101061486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-33}) \)$)$

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.a_cg	$2$	(not in LMFDB)
2.37.i_dm	$2$	(not in LMFDB)
2.37.e_av	$3$	(not in LMFDB)