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

• Label: 2.41.as_fz
• 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:	$1 - 18 x + 155 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.125194952327$, $\pm0.339939988779$
Angle rank:	$2$ (numerical)
Number field:	4.0.194112.5
Galois group:	$D_{4}$
Jacobians:	$12$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1081$	$2803033$	$4772467984$	$7989005641257$	$13422156852785881$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1668$	$69246$	$2827204$	$115851864$	$4750065294$	$194754764760$	$7984934476420$	$327382003307790$	$13422659545009668$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=5x^6+39x^5+19x^4+18x^3+32x^2+x+39$
• $y^2=12x^6+40x^5+38x^4+32x^3+9x^2+27$
• $y^2=24x^6+14x^5+40x^4+19x^3+32x^2+16x+26$
• $y^2=19x^6+36x^5+21x^4+2x^3+3x+20$
• $y^2=8x^6+14x^5+39x^4+33x^3+29x^2+5x+6$
• $y^2=19x^6+27x^5+15x^4+3x^3+4x^2+40x+17$
• $y^2=24x^6+26x^5+38x^4+36x^3+32x^2+5x+28$
• $y^2=27x^6+15x^5+34x^4+32x^3+14x^2+31x+26$
• $y^2=6x^6+17x^5+3x^4+32x^3+16x^2+28x+38$
• $y^2=5x^6+25x^5+19x^4+18x^3+13x^2+9$
• $y^2=15x^6+5x^5+19x^4+40x^3+18x^2+6x+23$
• $y^2=14x^6+3x^4+9x^3+36x^2+24x+11$

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 4.0.194112.5.

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_fz	$2$	(not in LMFDB)