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

• Label: 2.41.ax_if
• 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:	$1 - 23 x + 213 x^{2} - 943 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0546404201888$, $\pm0.199089310397$
Angle rank:	$2$ (numerical)
Number field:	4.0.6725.1
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$929$	$2657869$	$4729562225$	$7985197966709$	$13423522599357184$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$1579$	$68623$	$2825859$	$115863654$	$4750146163$	$194754244879$	$7984922497059$	$327381904738843$	$13422659112195054$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=21x^6+28x^5+18x^4+30x^3+13x^2+16x+6$

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 4.0.6725.1.

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.x_if	$2$	(not in LMFDB)