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

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

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 80 x^{2} + 1681 x^{4}$
Frobenius angles:	$\pm0.0352232874773$, $\pm0.964776712523$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{8})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	17
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1602$	$2566404$	$4749995682$	$7967770780176$	$13422659206408002$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$1522$	$68922$	$2819686$	$115856202$	$4749887122$	$194754273882$	$7984918073278$	$327381934393962$	$13422659102663602$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{2}}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).

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

The base change of $A$ to $\F_{41^{2}}$ is 1.1681.adc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_dc	$4$	(not in LMFDB)
2.41.au_ha	$8$	(not in LMFDB)
2.41.as_gg	$8$	(not in LMFDB)