Abelian variety isogeny class 2.61.az_ks over $\F_{61}$

• Label: 2.61.az_ks
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x + 61 x^{2} )( 1 - 12 x + 61 x^{2} )$
	$1 - 25 x + 278 x^{2} - 1525 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.187058313935$, $\pm0.221142061624$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	9

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2450$	$13597500$	$51668451800$	$191876171760000$	$713437405691386250$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3653$	$227632$	$13858033$	$844708177$	$51521030138$	$3142744301797$	$191707292582593$	$11694145758752992$	$713342908781301773$

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 $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.an $\times$ 1.61.am and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.61.an : \(\Q(\sqrt{-3}) \).
• 1.61.am : \(\Q(\sqrt{-1}) \).

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
2.61.ab_abi	$2$	(not in LMFDB)
2.61.b_abi	$2$	(not in LMFDB)
2.61.z_ks	$2$	(not in LMFDB)
2.61.an_fe	$3$	(not in LMFDB)
2.61.c_abu	$3$	(not in LMFDB)