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

• Label: 2.59.ay_kb
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x + 59 x^{2} )( 1 - 11 x + 59 x^{2} )$
	$1 - 24 x + 261 x^{2} - 1416 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.178868127011$, $\pm0.245953251861$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	8
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2303$	$11936449$	$42328882064$	$146967587994745$	$511181597534110223$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3428$	$206100$	$12128676$	$715014996$	$42180964886$	$2488651677324$	$146830418711236$	$8662995609906060$	$511116751993189028$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=58 x^6+47 x^5+27 x^4+50 x^3+45 x^2+6 x+50$
• $y^2=8 x^6+37 x^5+6 x^4+4 x^3+8 x^2+33 x+43$
• $y^2=8 x^6+53 x^5+24 x^4+49 x^3+37 x^2+22 x+40$
• $y^2=6 x^6+32 x^5+42 x^4+11 x^3+39 x^2+30 x+50$
• $y^2=40 x^6+37 x^5+54 x^4+29 x^3+23 x^2+23 x+10$
• $y^2=44 x^6+15 x^5+x^4+39 x^3+26 x^2+51 x+31$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.an $\times$ 1.59.al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.59.an : \(\Q(\sqrt{-67}) \).
• 1.59.al : \(\Q(\sqrt{-115}) \).

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.ac_az	$2$	(not in LMFDB)
2.59.c_az	$2$	(not in LMFDB)
2.59.y_kb	$2$	(not in LMFDB)