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

• Label: 2.61.ay_kf
• 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:	$( 1 - 13 x + 61 x^{2} )( 1 - 11 x + 61 x^{2} )$
	$1 - 24 x + 265 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.187058313935$, $\pm0.251304563322$
Angle rank:	$2$ (numerical)
Jacobians:	$14$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2499$	$13682025$	$51717064896$	$191882861829225$	$713424990133573179$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$3676$	$227846$	$13858516$	$844693478$	$51520784038$	$3142742256398$	$191707285627876$	$11694145827742526$	$713342910180602476$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=22x^6+14x^5+49x^4+16x^3+49x^2+14x+22$
• $y^2=29x^6+48x^5+2x^4+54x^3+2x^2+48x+29$
• $y^2=55x^6+45x^5+30x^4+16x^3+30x^2+45x+55$
• $y^2=28x^6+12x^5+14x^4+23x^3+14x^2+12x+28$
• $y^2=44x^6+27x^5+4x^4+27x^3+4x^2+27x+44$
• $y^2=27x^6+46x^5+12x^4+34x^3+12x^2+46x+27$
• $y^2=21x^6+41x^5+7x^4+11x^3+7x^2+41x+21$
• $y^2=48x^6+28x^5+20x^4+20x^3+20x^2+28x+48$
• $y^2=53x^6+35x^5+19x^4+11x^3+19x^2+35x+53$
• $y^2=22x^6+44x^5+5x^4+45x^3+5x^2+44x+22$
• $y^2=38x^6+27x^5+54x^4+58x^3+54x^2+27x+38$
• $y^2=11x^6+45x^5+12x^4+37x^3+12x^2+45x+11$
• $y^2=11x^6+14x^5+60x^4+7x^3+60x^2+14x+11$
• $y^2=38x^6+57x^5+4x^4+x^3+4x^2+57x+38$

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.al 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.al : \(\Q(\sqrt{-123}) \).

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.ac_av	$2$	(not in LMFDB)
2.61.c_av	$2$	(not in LMFDB)
2.61.y_kf	$2$	(not in LMFDB)
2.61.am_fd	$3$	(not in LMFDB)
2.61.d_abg	$3$	(not in LMFDB)