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

• Label: 2.59.abe_nf
• 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:	$( 1 - 15 x + 59 x^{2} )^{2}$
	$1 - 30 x + 343 x^{2} - 1770 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0692665268586$, $\pm0.0692665268586$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2025$	$11390625$	$41885715600$	$146721740765625$	$511081259355275625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$3268$	$203940$	$12108388$	$714874650$	$42180318358$	$2488651184910$	$146830445807428$	$8662995959396220$	$511116754927777348$

Jacobians and polarizations

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

• $y^2=47 x^6+8 x^5+52 x^4+17 x^3+13 x^2+30 x+33$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ap^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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
2.59.a_aed	$2$	(not in LMFDB)
2.59.be_nf	$2$	(not in LMFDB)
2.59.p_gk	$3$	(not in LMFDB)