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

• Label: 2.37.s_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:	$\pm0.765077740875$, $\pm0.765077740875$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2209$	$1857769$	$2538547456$	$3522569676201$	$4806613819084009$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$1356$	$50114$	$1879540$	$69315536$	$2565783222$	$94932417440$	$3512472489124$	$129961782486938$	$4808584245888636$

Jacobians and polarizations

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

• $y^2=3 x^6+3 x^5+10 x^4+22 x^3+24 x^2+17 x+12$
• $y^2=15 x^6+10 x^5+11 x^4+28 x^3+5 x^2+29 x+29$
• $y^2=21 x^6+16 x^5+26 x^4+10 x^3+30 x^2+34 x+12$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.j^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.as_fz	$2$	(not in LMFDB)
2.37.a_ah	$2$	(not in LMFDB)
2.37.aj_bs	$3$	(not in LMFDB)