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

• Label: 2.61.aba_lb
• 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:	$( 1 - 15 x + 61 x^{2} )( 1 - 11 x + 61 x^{2} )$
	$1 - 26 x + 287 x^{2} - 1586 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0900194921159$, $\pm0.251304563322$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2397$	$13473537$	$51532201728$	$191766494252025$	$713369242431934077$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3620$	$227034$	$13850116$	$844627476$	$51520420262$	$3142741667556$	$191707302982276$	$11694146124592674$	$713342913338930180$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=27x^6+2x^5+52x^4+53x^3+3x^2+7x+60$
• $y^2=x^6+24x^5+53x^4+22x^3+25x^2+49x+24$
• $y^2=45x^6+22x^5+23x^4+23x^3+11x^2+56x+15$
• $y^2=35x^6+23x^5+39x^4+22x^3+13x^2+54x+21$
• $y^2=38x^6+40x^5+12x^4+23x^3+49x^2+40x+23$
• $y^2=5x^6+54x^5+31x^4+29x^3+30x+24$
• $y^2=51x^6+29x^5+19x^4+23x^3+46x^2+22x+50$
• $y^2=32x^6+42x^5+39x^4+38x^3+45x^2+57x+30$
• $y^2=44x^6+45x^5+35x^4+17x^3+58x^2+6x+57$
• $y^2=40x^6+13x^5+11x^4+54x^3+45x^2+33x+54$
• $y^2=24x^6+41x^5+52x^4+35x^3+58x^2+52x+28$
• $y^2=53x^6+33x^5+43x^4+13x^3+55x^2+24x+11$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ap $\times$ 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.ap : \(\Q(\sqrt{-19}) \).
• 1.61.al : \(\Q(\sqrt{-123}) \).

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.ae_abr	$2$	(not in LMFDB)
2.61.e_abr	$2$	(not in LMFDB)
2.61.ba_lb	$2$	(not in LMFDB)