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

• Label: 2.41.as_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:	$\pm0.251940962052$, $\pm0.251940962052$
Angle rank:	$1$ (numerical)
Jacobians:	$15$
Cyclic group of points:	no
Non-cyclic primes:	$3, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1089$	$2832489$	$4802490000$	$8003936949129$	$13426077749128209$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1684$	$69678$	$2832484$	$115885704$	$4750094158$	$194752973544$	$7984913939524$	$327381886101438$	$13422659338393204$

Jacobians and polarizations

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

• $y^2=29 x^6+27 x^5+25 x^4+35 x^3+9 x^2+26 x+27$
• $y^2=6 x^6+18 x^5+6 x^4+33 x^3+17 x^2+x+26$
• $y^2=18 x^6+25 x^5+6 x^4+25 x^3+24 x^2+31 x+4$
• $y^2=20 x^6+28 x^5+29 x^4+5 x^3+14 x^2+29 x+10$
• $y^2=4 x^6+39 x^5+26 x^4+36 x^3+35 x^2+21 x+16$
• $y^2=39 x^6+3 x^5+10 x^4+34 x^3+9 x^2+34 x+37$
• $y^2=33 x^6+21 x^5+30 x^4+23 x^3+6 x^2+32 x+37$
• $y^2=39 x^6+30 x^5+35 x^4+38 x^3+30 x^2+12 x+4$
• $y^2=14 x^6+27 x^5+34 x^4+3 x^3+21 x^2+12 x+31$
• $y^2=30 x^6+20 x^5+33 x^4+10 x^3+20 x^2+2 x+13$
• $y^2=28 x^6+36 x^5+16 x^4+37 x^3+18 x^2+2 x+12$
• $y^2=38 x^6+29 x^5+x^4+24 x^3+23 x^2+11 x+25$
• $y^2=18 x^6+16 x^5+34 x^4+18 x^3+20 x^2+32 x+32$
• $y^2=11 x^6+21 x^5+40 x^4+26 x^3+37 x^2+8 x+7$
• $y^2=15 x^6+28 x^5+35 x^4+12 x^3+17 x^2+38 x+17$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{41}$

The isogeny class factors as 1.41.aj^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.a_b	$2$	(not in LMFDB)
2.41.s_gh	$2$	(not in LMFDB)
2.41.j_bo	$3$	(not in LMFDB)