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

• Label: 2.59.ax_jq
• 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:	$( 1 - 12 x + 59 x^{2} )( 1 - 11 x + 59 x^{2} )$
	$1 - 23 x + 250 x^{2} - 1357 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.214641822575$, $\pm0.245953251861$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2352$	$12023424$	$42389032896$	$146990927301120$	$511182677845377552$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3453$	$206392$	$12130601$	$715016507$	$42180818886$	$2488649436233$	$146830398215761$	$8662995490968328$	$511116751838122653$

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.am $\times$ 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.am : \(\Q(\sqrt{-23}) \).
• 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.ab_ao	$2$	(not in LMFDB)
2.59.b_ao	$2$	(not in LMFDB)
2.59.x_jq	$2$	(not in LMFDB)