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

• Label: 2.61.aba_ld
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 26 x + 289 x^{2} - 1586 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.125915170111$, $\pm0.234017939091$
Angle rank:	$2$ (numerical)
Number field:	4.0.254528.3
Galois group:	$D_{4}$
Jacobians:	$4$
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2399$	$13489577$	$51567838844$	$191809495935353$	$713404598749519959$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3624$	$227190$	$13853220$	$844669336$	$51520835550$	$3142744647624$	$191707315193988$	$11694146086080174$	$713342912009551624$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=10 x^6+26 x^5+20 x^4+5 x^3+11 x^2+11 x+32$
• $y^2=45 x^6+52 x^5+7 x^4+60 x^3+49 x^2+9 x+28$
• $y^2=23 x^6+21 x^5+35 x^4+27 x^3+32 x^2+17 x+54$
• $y^2=27 x^6+41 x^4+2 x^3+55 x^2+26 x+9$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is 4.0.254528.3.

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
2.61.ba_ld	$2$	(not in LMFDB)