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

• Label: 2.41.as_fw
• 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:	$1 - 18 x + 152 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.0883036955963$, $\pm0.353631113310$
Angle rank:	$2$ (numerical)
Number field:	4.0.38720.3
Galois group:	$D_{4}$
Jacobians:	$22$
Isomorphism classes:	22

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1078$	$2792020$	$4761225238$	$7983222786000$	$13420342712719558$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1662$	$69084$	$2825158$	$115836204$	$4749989262$	$194754525864$	$7984933498558$	$327381993703944$	$13422659500021902$

Jacobians and polarizations

This isogeny class contains the Jacobians of 22 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=30x^6+16x^5+19x^4+32x^3+12x^2+6x+14$
• $y^2=17x^6+6x^5+18x^4+32x^3+39x^2+12x+16$
• $y^2=24x^6+6x^5+8x^4+16x^3+10x^2+27x+20$
• $y^2=30x^6+15x^5+9x^4+23x^3+18x^2+27x+19$
• $y^2=24x^6+34x^5+31x^4+24x^3+33x^2+5x+16$
• $y^2=2x^6+40x^5+x^4+33x^3+39x^2+7x+15$
• $y^2=24x^6+12x^5+18x^4+28x^3+4x^2+17x+2$
• $y^2=24x^6+36x^5+32x^4+9x^3+36x^2+38x+27$
• $y^2=35x^6+22x^5+28x^4+22x^3+17x^2+29x+15$
• $y^2=11x^6+15x^5+22x^4+35x^3+16x^2+33x+28$
• $y^2=19x^6+35x^5+17x^4+27x^3+13x^2+20x+14$
• $y^2=11x^6+18x^5+40x^4+11x^3+10x^2+8x+10$
• $y^2=27x^6+5x^5+37x^4+5x^3+7x^2+2x+34$
• $y^2=34x^6+21x^5+33x^4+27x^3+19x^2+18x+36$
• $y^2=10x^6+3x^5+34x^4+19x^3+28x^2+37x+17$
• $y^2=14x^6+21x^5+8x^4+36x^3+30x^2+30x$
• $y^2=9x^5+35x^4+35x^3+17x^2+27x+34$
• $y^2=21x^6+9x^5+15x^4+11x^3+10x^2+14x+13$
• $y^2=23x^6+39x^5+29x^4+19x^3+36x^2+12x+29$
• $y^2=28x^6+27x^5+8x^4+38x^3+20x^2+2x+3$
• $y^2=10x^6+39x^5+16x^4+6x^3+33x^2+27x+24$
• $y^2=30x^6+27x^5+10x^4+28x^3+34x^2+29x+13$

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 4.0.38720.3.

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_fw	$2$	(not in LMFDB)