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

• Label: 2.59.aj_fi
• 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: no

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 59 x^{2} )( 1 - 4 x + 59 x^{2} )$
	$1 - 9 x + 138 x^{2} - 531 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.394476720982$, $\pm0.416152878126$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3080$	$12812800$	$42469787360$	$146768317696000$	$511041662554669400$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$51$	$3677$	$206784$	$12112233$	$714819261$	$42180362822$	$2488656831999$	$146830471380433$	$8662995659664576$	$511116750642601277$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.af $\times$ 1.59.ae 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.af : \(\Q(\sqrt{-211}) \).
• 1.59.ae : \(\Q(\sqrt{-55}) \).

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.ab_du	$2$	(not in LMFDB)
2.59.b_du	$2$	(not in LMFDB)
2.59.j_fi	$2$	(not in LMFDB)