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

• Label: 2.41.av_hf
• 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:	$1 - 21 x + 187 x^{2} - 861 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0153876165359$, $\pm0.278522863005$
Angle rank:	$2$ (numerical)
Number field:	4.0.16317.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$987$	$2715237$	$4745662803$	$7984493803125$	$13421063357891952$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1615$	$68859$	$2825611$	$115842426$	$4749901567$	$194752569711$	$7984915734451$	$327381902340129$	$13422659285930830$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=20x^6+11x^5+27x^4+29x^3+20x^2+18x+24$
• $y^2=20x^6+5x^5+33x^4+37x^3+21x^2+29x+7$

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 4.0.16317.1.

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_hf	$2$	(not in LMFDB)