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

• Label: 2.41.av_hh
• 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:	$1 - 21 x + 189 x^{2} - 861 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0895520221991$, $\pm0.262353290689$
Angle rank:	$2$ (numerical)
Number field:	4.0.188773.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$989$	$2722717$	$4754440469$	$7990218721333$	$13423642423672784$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1619$	$68985$	$2827635$	$115864686$	$4750088555$	$194753847729$	$7984923400963$	$327381948777429$	$13422659617179374$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=40x^6+33x^5+31x^4+16x^3+28x^2+39x+11$
• $y^2=13x^6+13x^5+15x^4+8x^3+30x^2+7x+14$
• $y^2=12x^6+36x^5+25x^4+8x^3+34x^2+10x+9$

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 4.0.188773.1.

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.v_hh	$2$	(not in LMFDB)