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

• Label: 2.37.as_fz
• 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:	$( 1 - 9 x + 37 x^{2} )^{2}$
	$1 - 18 x + 155 x^{2} - 666 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.234922259125$, $\pm0.234922259125$
Angle rank:	$1$ (numerical)
Jacobians:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$841$	$1857769$	$2593253776$	$3522569676201$	$4810555607032561$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$1356$	$51194$	$1879540$	$69372380$	$2565783222$	$94931336828$	$3512472489124$	$129961697103218$	$4808584245888636$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=20x^6+20x^5+5x^4+11x^3+12x^2+27x+6$
• $y^2=30x^6+20x^5+22x^4+19x^3+10x^2+21x+21$
• $y^2=5x^6+32x^5+15x^4+20x^3+23x^2+31x+24$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.aj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

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_ah	$2$	(not in LMFDB)
2.37.s_fz	$2$	(not in LMFDB)
2.37.j_bs	$3$	(not in LMFDB)