Abelian variety isogeny class 2.59.az_kj over $\F_{59}$

• Label: 2.59.az_kj
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 - 25 x + 269 x^{2} - 1475 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0870584688725$, $\pm0.268631548773$
Angle rank:	$2$ (numerical)
Number field:	4.0.1000629.1
Galois group:	$D_{4}$
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2251$	$11820001$	$42206157709$	$146873784005869$	$511127120859737776$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$3395$	$205505$	$12120939$	$714938800$	$42180421151$	$2488649612995$	$146830430004739$	$8662995915787565$	$511116755403884150$

Jacobians and polarizations

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

• $y^2=52 x^6+6 x^5+7 x^4+46 x^3+38 x^2+11 x+55$
• $y^2=50 x^6+31 x^5+35 x^4+36 x^3+5 x^2+51 x+21$
• $y^2=31 x^6+48 x^5+50 x^4+11 x^3+14 x^2+28 x+1$
• $y^2=16 x^6+41 x^5+58 x^4+12 x^3+29 x^2+10 x+7$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is 4.0.1000629.1.

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
2.59.z_kj	$2$	(not in LMFDB)