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

• Label: 2.41.s_gh
• 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:	$( 1 + 9 x + 41 x^{2} )^{2}$
	$1 + 18 x + 163 x^{2} + 738 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.748059037948$, $\pm0.748059037948$
Angle rank:	$1$ (numerical)
Jacobians:	$15$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2601$	$2832489$	$4698279936$	$8003936949129$	$13419241769785401$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$1684$	$68166$	$2832484$	$115826700$	$4750094158$	$194755574220$	$7984913939524$	$327381982686486$	$13422659338393204$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

• $y^2=10 x^6+39 x^5+27 x^4+5 x^3+13 x^2+33 x+39$
• $y^2=36 x^6+26 x^5+36 x^4+34 x^3+20 x^2+6 x+33$
• $y^2=26 x^6+27 x^5+36 x^4+27 x^3+21 x^2+22 x+24$
• $y^2=17 x^6+32 x^5+39 x^4+35 x^3+16 x^2+39 x+29$
• $y^2=28 x^6+27 x^5+18 x^4+6 x^3+40 x^2+24 x+30$
• $y^2=27 x^6+21 x^5+29 x^4+33 x^3+22 x^2+33 x+13$
• $y^2=26 x^6+24 x^5+5 x^4+38 x^3+x^2+19 x+13$
• $y^2=27 x^6+5 x^5+40 x^4+20 x^3+5 x^2+2 x+28$
• $y^2=16 x^6+25 x^5+33 x^4+21 x^3+24 x^2+2 x+12$
• $y^2=5 x^6+17 x^5+26 x^4+29 x^3+17 x^2+14 x+9$
• $y^2=32 x^6+6 x^5+30 x^4+13 x^3+3 x^2+14 x+2$
• $y^2=23 x^6+10 x^5+6 x^4+21 x^3+15 x^2+25 x+27$
• $y^2=26 x^6+14 x^5+40 x^4+26 x^3+38 x^2+28 x+28$
• $y^2=25 x^6+3 x^5+35 x^4+33 x^3+17 x^2+7 x+1$
• $y^2=8 x^6+4 x^5+5 x^4+31 x^3+20 x^2+23 x+20$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.j^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-83}) \)$)$

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.as_gh	$2$	(not in LMFDB)
2.41.a_b	$2$	(not in LMFDB)
2.41.aj_bo	$3$	(not in LMFDB)