Abelian variety isogeny class 2.41.as_ge over $\F_{41}$

• Label: 2.41.as_ge
• Base field: $\F_{41}$
• 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:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 18 x + 160 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.183705984994$, $\pm0.307898514763$
Angle rank:	$2$ (numerical)
Number field:	4.0.781632.1
Galois group:	$D_{4}$
Jacobians:	$16$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1086$	$2821428$	$4791224574$	$7998421094352$	$13424763666724206$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1678$	$69516$	$2830534$	$115874364$	$4750112734$	$194754044040$	$7984925115070$	$327381944852520$	$13422659332756318$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=13x^6+36x^5+26x^4+21x^3+15x^2+8x+34$
• $y^2=33x^6+26x^5+12x^4+5x^3+11x^2+35x+14$
• $y^2=15x^6+30x^5+29x^4+26x^3+12x^2+23x+7$
• $y^2=24x^6+16x^5+6x^4+29x^3+33x^2+37x+36$
• $y^2=28x^6+15x^5+26x^4+21x^3+10x^2+15x+28$
• $y^2=34x^6+29x^5+26x^4+37x^3+40x^2+38x+38$
• $y^2=15x^6+37x^5+4x^4+16x^3+5x^2+7x+25$
• $y^2=6x^6+9x^5+35x^4+32x^3+21x^2+4x+28$
• $y^2=34x^6+3x^5+15x^4+25x^3+20x^2+24x+33$
• $y^2=37x^6+10x^5+18x^4+17x^3+20x^2+13x+11$
• $y^2=21x^6+27x^5+6x^4+40x^3+35x^2+13x+33$
• $y^2=13x^6+11x^5+x^4+22x^3+11x^2+2x+7$
• $y^2=24x^6+13x^5+14x^4+31x^3+39x^2+2x+12$
• $y^2=16x^6+34x^5+x^4+5x^3+5x^2+29x+9$
• $y^2=13x^6+18x^5+19x^4+8x^3+27x^2+40$
• $y^2=13x^6+x^5+31x^3+40x^2+28x+37$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is 4.0.781632.1.

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
2.41.s_ge	$2$	(not in LMFDB)