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

• Label: 2.41.v_hj
• 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:	$1 + 21 x + 191 x^{2} + 861 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.761697326518$, $\pm0.861796775820$
Angle rank:	$2$ (numerical)
Number field:	4.0.55125.1
Galois group:	$C_4$
Jacobians:	$4$
Isomorphism classes:	4
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2755$	$2730205$	$4737170155$	$7995899509605$	$13419195193750000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$1623$	$68733$	$2829643$	$115826298$	$4750252863$	$194753840673$	$7984925163283$	$327381940825443$	$13422659310827598$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=5 x^6+17 x^5+33 x^4+34 x^3+25 x^2+8 x+26$
• $y^2=35 x^6+37 x^5+7 x^4+32 x^3+3 x^2+9 x+1$
• $y^2=40 x^6+32 x^5+14 x^4+10 x^3+x^2+19 x+2$
• $y^2=5 x^6+25 x^5+30 x^4+11 x^3+8 x^2+19 x+9$

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 4.0.55125.1.

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.av_hj	$2$	(not in LMFDB)