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

• Label: 2.37.ay_ik
• 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: no

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$( 1 - 12 x + 37 x^{2} )^{2}$
	$1 - 24 x + 218 x^{2} - 888 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.0525684567113$, $\pm0.0525684567113$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$676$	$1690000$	$2525866564$	$3504384000000$	$4807018574667556$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$1230$	$49862$	$1869838$	$69321374$	$2565615390$	$94931380502$	$3512477602078$	$129961735948334$	$4808584394775150$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_acs	$2$	(not in LMFDB)
2.37.y_ik	$2$	(not in LMFDB)
2.37.m_ed	$3$	(not in LMFDB)