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

• Label: 2.59.ae_abr
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 - 4 x - 43 x^{2} - 236 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0828195447922$, $\pm0.749486211459$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-55})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	120
Cyclic group of points:	yes

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3199$	$11769121$	$41916829696$	$146872145294329$	$511079747923763599$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$3380$	$204092$	$12120804$	$714872536$	$42180525686$	$2488654522984$	$146830425216964$	$8662996078059428$	$511116754550304500$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{3}}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-55})\).

Endomorphism algebra over $\overline{\F}_{59}$

The base change of $A$ to $\F_{59^{3}}$ is 1.205379.ayu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-55}) \)$)$

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.e_abr	$2$	(not in LMFDB)
2.59.i_fe	$3$	(not in LMFDB)
2.59.ai_fe	$6$	(not in LMFDB)