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

• Label: 2.61.ay_kg
• 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: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 - 12 x + 61 x^{2} )^{2}$
	$1 - 24 x + 266 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.221142061624$, $\pm0.221142061624$
Angle rank:	$1$ (numerical)
Jacobians:	$9$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2500$	$13690000$	$51733502500$	$191900067840000$	$713435734126562500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3678$	$227918$	$13859758$	$844706198$	$51520844238$	$3142741770878$	$191707271553118$	$11694145660477958$	$713342909002702398$

Jacobians and polarizations

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

• $y^2=4x^6+59x^5+17x^4+13x^3+15x^2+59x+30$
• $y^2=24x^6+29x^5+51x^4+60x^3+x^2+3x+10$
• $y^2=23x^6+5x^5+55x^4+34x^3+26x^2+60x+24$
• $y^2=31x^6+23x^5+38x^4+2x^3+4x^2+54x+29$
• $y^2=28x^6+59x^4+59x^2+28$
• $y^2=18x^6+13x^5+41x^4+24x^3+19x^2+25x+7$
• $y^2=54x^6+57x^5+23x^4+34x^3+40x^2+47x+56$
• $y^2=57x^6+37x^5+x^4+23x^3+46x^2+29x+19$
• $y^2=50x^6+6x^5+4x^4+33x^3+9x^2+38x+11$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_aw	$2$	(not in LMFDB)
2.61.y_kg	$2$	(not in LMFDB)
2.61.m_df	$3$	(not in LMFDB)