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

• Label: 2.41.as_gd
• 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:	$( 1 - 11 x + 41 x^{2} )( 1 - 7 x + 41 x^{2} )$
	$1 - 18 x + 159 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.171113726078$, $\pm0.315918729109$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1085$	$2817745$	$4787471360$	$7996560250105$	$13424283974457125$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1676$	$69462$	$2829876$	$115870224$	$4750111118$	$194754296544$	$7984927976356$	$327381961744422$	$13422659370816476$

Jacobians and polarizations

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

• $y^2=37x^6+31x^5+40x^4+11x^3+21x^2+18x+21$
• $y^2=24x^6+26x^5+7x^4+11x^3+24x^2+27x+3$
• $y^2=34x^6+14x^5+25x^4+20x^3+32x^2+15x+15$
• $y^2=24x^6+15x^5+27x^4+29x^3+34x^2+14x+3$
• $y^2=28x^6+39x^5+27x^4+5x^3+16x^2+5x+34$
• $y^2=17x^6+16x^5+29x^4+30x^3+26x^2+25x+30$
• $y^2=12x^6+24x^5+7x^4+35x^3+14x^2+14x+14$
• $y^2=22x^6+4x^4+24x^3+9x^2+16x+4$
• $y^2=34x^6+34x^5+22x^4+10x^3+40x^2+34x+27$
• $y^2=14x^6+35x^5+32x^4+14x^3+30x^2+15x+40$
• $y^2=2x^6+2x^5+9x^4+21x^3+16x^2+10x+13$
• $y^2=22x^6+38x^5+21x^4+21x^3+12x^2+24x+35$

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 $\times$ 1.41.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.41.al : \(\Q(\sqrt{-43}) \).
• 1.41.ah : \(\Q(\sqrt{-115}) \).

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.ae_f	$2$	(not in LMFDB)
2.41.e_f	$2$	(not in LMFDB)
2.41.s_gd	$2$	(not in LMFDB)