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

• Label: 2.37.at_gd
• Base field: $\F_{37}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$1 - 19 x + 159 x^{2} - 703 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.0791576389440$, $\pm0.298120274749$
Angle rank:	$2$ (numerical)
Number field:	4.0.379701.1
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$807$	$1816557$	$2570499171$	$3513906079989$	$4808247219460752$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$1327$	$50749$	$1874923$	$69339094$	$2565638647$	$94931387941$	$3512479652371$	$129961766308723$	$4808584620868582$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=9x^6+20x^5+26x^4+20x^3+6x^2+32x+13$
• $y^2=23x^6+11x^4+36x^3+34x^2+26x+17$
• $y^2=4x^6+26x^5+13x^4+20x^3+5x^2+15x+35$
• $y^2=18x^6+32x^5+9x^4+6x^3+7x^2+32x+36$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is 4.0.379701.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.37.t_gd	$2$	(not in LMFDB)