Abelian variety isogeny class 1.41.ae over $\F_{41}$

• Label: 1.41.ae
• Base field: $\F_{41}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{41}$
Dimension:	$1$
L-polynomial:	$1 - 4 x + 41 x^{2}$
Frobenius angles:	$\pm0.398884665197$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-37}) \)
Galois group:	$C_2$
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:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$38$	$1748$	$69350$	$2824768$	$115834678$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$1748$	$69350$	$2824768$	$115834678$	$4750058900$	$194754974998$	$7984929892608$	$327381924302150$	$13422659078582228$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):

• $y^2=x^3+37 x+29$
• $y^2=x^3+37 x+37$

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 \(\Q(\sqrt{-37}) \).

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
1.41.e	$2$	(not in LMFDB)