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

• Label: 2.41.ay_is
• 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:	$( 1 - 12 x + 41 x^{2} )^{2}$
	$1 - 24 x + 226 x^{2} - 984 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.113551764296$, $\pm0.113551764296$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$900$	$2624400$	$4715568900$	$7982207078400$	$13423713390562500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$1558$	$68418$	$2824798$	$115865298$	$4750252918$	$194755685058$	$7984936067518$	$327382006596498$	$13422659732208598$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=8 x^6+17 x^4+30 x^3+35 x^2+36$
• $y^2=29 x^6+26 x^4+26 x^2+29$
• $y^2=27 x^6+10 x^5+9 x^4+19 x^3+x^2+31 x+38$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

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_ack	$2$	(not in LMFDB)
2.41.y_is	$2$	(not in LMFDB)
2.41.m_dz	$3$	(not in LMFDB)