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

• Label: 2.41.aw_ht
• 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:	$1 - 22 x + 201 x^{2} - 902 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0789642873101$, $\pm0.230762790725$
Angle rank:	$2$ (numerical)
Number field:	4.0.45632.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$959$	$2691913$	$4744061756$	$7988954416793$	$13424058719930719$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$1600$	$68834$	$2827188$	$115868280$	$4750141798$	$194754106616$	$7984922497764$	$327381921496946$	$13422659361415440$

Jacobians and polarizations

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

• $y^2=19x^6+22x^5+12x^4+31x^3+4x^2+38x+11$
• $y^2=11x^6+21x^5+4x^4+21x^3+18x^2+31x+6$

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 4.0.45632.1.

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.w_ht	$2$	(not in LMFDB)