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

• Label: 2.59.abb_ll
• 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:	$1 - 27 x + 297 x^{2} - 1593 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0280281080988$, $\pm0.224500357009$
Angle rank:	$2$ (numerical)
Number field:	4.0.30589.1
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2159$	$11656441$	$42097354337$	$146831303228221$	$511117601662791824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$3347$	$204975$	$12117435$	$714925488$	$42180368807$	$2488648207533$	$146830399711939$	$8662995502197255$	$511116751363394102$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=18 x^6+46 x^5+47 x^4+53 x^3+43 x^2+6 x+41$

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 4.0.30589.1.

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.bb_ll	$2$	(not in LMFDB)