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

• Label: 2.41.ac_abl
• Base field: $\F_{41}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 2 x - 37 x^{2} - 82 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.116750683713$, $\pm0.783417350380$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-10})\)
Galois group:	$C_2^2$
Jacobians:	$0$

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1561$	$2698969$	$4717491856$	$7992621536809$	$13420898179414201$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$1604$	$68446$	$2828484$	$115841000$	$4750266638$	$194755059560$	$7984926986884$	$327382005850606$	$13422659309540804$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{3}}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-10})\).

Endomorphism algebra over $\overline{\F}_{41}$

The base change of $A$ to $\F_{41^{3}}$ is 1.68921.aje^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.41.c_abl	$2$	(not in LMFDB)
2.41.e_di	$3$	(not in LMFDB)
2.41.ae_di	$6$	(not in LMFDB)