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

• Label: 2.61.aba_la
• 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:	$1 - 26 x + 286 x^{2} - 1586 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0707655388053$, $\pm0.258010398137$
Angle rank:	$2$ (numerical)
Number field:	4.0.38000.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2396$	$13465520$	$51514385756$	$191744911281920$	$713351235086432316$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$3618$	$226956$	$13848558$	$844606156$	$51520199538$	$3142739876676$	$191707291576158$	$11694146068320276$	$713342913108960898$

Jacobians and polarizations

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

• $y^2=2x^6+46x^5+6x^4+37x^3+25x^2+39x+35$
• $y^2=24x^6+26x^5+25x^4+30x^3+32x^2+7x+11$
• $y^2=51x^6+10x^5+59x^4+56x^3+46x^2+16x+24$
• $y^2=17x^6+48x^5+x^4+22x^3+13x^2+16x+37$
• $y^2=53x^6+28x^5+30x^4+12x^3+54x^2+36x+4$
• $y^2=x^6+6x^5+42x^4+45x^3+45x^2+13x+51$
• $y^2=51x^6+25x^5+58x^4+29x^3+37x^2+18x+16$
• $y^2=28x^6+27x^5+48x^4+30x^3+28x^2+32x+44$
• $y^2=8x^6+17x^5+55x^4+32x^3+46x^2+7x+59$
• $y^2=17x^6+28x^5+54x^4+8x^3+46x^2+6x+54$
• $y^2=3x^6+23x^5+20x^4+38x^3+53x^2+8x+28$
• $y^2=44x^6+19x^5+36x^4+39x^3+34x^2+9x+19$

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 4.0.38000.1.

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.ba_la	$2$	(not in LMFDB)