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

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

Invariants

Base field:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 - 18 x + 153 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.101373436493$, $\pm0.349335512379$
Angle rank:	$2$ (numerical)
Number field:	4.0.3342400.2
Galois group:	$D_{4}$
Jacobians:	$6$
Isomorphism classes:	12

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1079$	$2795689$	$4764971900$	$7985161545049$	$13420968266033159$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1664$	$69138$	$2825844$	$115841604$	$4750018598$	$194754663204$	$7984934429284$	$327382001764578$	$13422659546485904$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=35x^6+x^5+4x^4+16x^3+9x^2+25x+14$
• $y^2=13x^6+29x^5+6x^4+31x^3+39x^2+37x+30$
• $y^2=38x^6+13x^5+40x^3+9x^2+9x+11$
• $y^2=14x^6+39x^5+39x^4+19x^3+26x^2+23x+12$
• $y^2=22x^6+39x^4+x^3+28x^2+21x+1$
• $y^2=24x^6+7x^5+26x^4+16x^3+5x^2+6x+28$

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 4.0.3342400.2.

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
2.41.s_fx	$2$	(not in LMFDB)