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

• Label: 2.61.az_kr
• 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:	$1 - 25 x + 277 x^{2} - 1525 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.162949397008$, $\pm0.240145470170$
Angle rank:	$2$ (numerical)
Number field:	4.0.167525.1
Galois group:	$D_{4}$
Jacobians:	$9$
Isomorphism classes:	9

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2449$	$13589501$	$51651322669$	$191856914855525$	$713423574578330704$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3651$	$227557$	$13856643$	$844691802$	$51520911051$	$3142744131697$	$191707302173283$	$11694145926427417$	$713342910479846806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):

• $y^2=57 x^6+29 x^5+12 x^4+55 x^3+13 x^2+4 x+30$
• $y^2=15 x^6+7 x^5+22 x^4+10 x^3+30 x^2+47 x+34$
• $y^2=35 x^6+10 x^5+33 x^4+19 x^3+51 x^2+41 x+50$
• $y^2=7 x^6+60 x^5+x^4+3 x^3+45 x^2+59 x+45$
• $y^2=33 x^6+44 x^5+45 x^4+3 x^3+15 x^2+22 x+51$
• $y^2=12 x^6+22 x^5+x^4+32 x^3+47 x^2+49 x+50$
• $y^2=7 x^6+60 x^5+7 x^4+8 x^3+44 x^2+18 x+18$
• $y^2=17 x^6+8 x^5+20 x^4+60 x^3+22 x^2+33 x+28$
• $y^2=50 x^6+29 x^5+25 x^4+18 x^3+45 x^2+8 x+43$

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 4.0.167525.1.

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.z_kr	$2$	(not in LMFDB)