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

• Label: 2.37.aw_hn
• 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:	$( 1 - 11 x + 37 x^{2} )^{2}$
	$1 - 22 x + 195 x^{2} - 814 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.140472200256$, $\pm0.140472200256$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$729$	$1750329$	$2554695936$	$3514466345481$	$4809956093699409$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$1276$	$50434$	$1875220$	$69363736$	$2565904822$	$94933107880$	$3512486390884$	$129961770564058$	$4808584454209036$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=11x^6+36x^5+31x^4+19x^3+5x^2+25x+1$
• $y^2=x^6+12x^3+26$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.al^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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_abv	$2$	(not in LMFDB)
2.37.w_hn	$2$	(not in LMFDB)
2.37.ak_cl	$3$	(not in LMFDB)
2.37.ab_abk	$3$	(not in LMFDB)
2.37.c_cx	$3$	(not in LMFDB)
2.37.l_dg	$3$	(not in LMFDB)
2.37.u_gs	$3$	(not in LMFDB)