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

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

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$( 1 - 11 x + 41 x^{2} )^{2}$
	$1 - 22 x + 203 x^{2} - 902 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.171113726078$, $\pm0.171113726078$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Cyclic group of points:	no
Non-cyclic primes:	$31$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$961$	$2699449$	$4753275136$	$7995338725609$	$13427143319553601$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$1604$	$68966$	$2829444$	$115894900$	$4750378958$	$194755709140$	$7984929753604$	$327381925317686$	$13422659024809604$

Jacobians and polarizations

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

• $y^2=2 x^6+30 x^5+35 x^4+40 x^3+29 x^2+38 x+16$
• $y^2=9 x^6+17 x^5+40 x^4+x^3+9 x^2+29 x+11$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.al^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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_abn	$2$	(not in LMFDB)
2.41.w_hv	$2$	(not in LMFDB)
2.41.l_dc	$3$	(not in LMFDB)