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

• Label: 2.41.ag_y
• 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:	$1 - 6 x + 24 x^{2} - 246 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.161890963527$, $\pm0.632696464529$
Angle rank:	$2$ (numerical)
Number field:	4.0.382965568.1
Galois group:	$D_{4}$
Jacobians:	$32$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1454$	$2846932$	$4714276574$	$7990096861648$	$13426892565798734$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$1694$	$68400$	$2827590$	$115892736$	$4750130990$	$194754884724$	$7984934166718$	$327381918525684$	$13422659105744174$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=15 x^6+7 x^5+10 x^4+10 x^3+15 x^2+32 x+7$
• $y^2=12 x^6+40 x^5+35 x^4+40 x^3+8 x^2+28 x+1$
• $y^2=37 x^6+28 x^5+17 x^4+5 x^3+33 x^2+19 x+15$
• $y^2=7 x^6+30 x^5+38 x^4+27 x^3+39 x^2+10 x+22$
• $y^2=7 x^6+7 x^5+6 x^4+16 x^3+25 x^2+6 x+34$
• $y^2=34 x^6+17 x^5+36 x^4+4 x^3+31 x^2+18 x+35$
• $y^2=26 x^6+30 x^5+40 x^4+4 x^3+30 x^2+24 x+15$
• $y^2=17 x^6+27 x^5+10 x^4+12 x^3+12 x^2+40 x+35$
• $y^2=15 x^6+3 x^5+9 x^4+10 x^3+18 x^2+12 x+11$
• $y^2=14 x^6+33 x^5+11 x^4+37 x^3+34 x^2+19 x+37$
• $y^2=7 x^6+x^5+15 x^4+23 x^3+8 x^2+19 x+24$
• $y^2=x^6+24 x^5+16 x^4+13 x^3+3 x^2+8 x+1$
• $y^2=5 x^6+34 x^5+10 x^4+10 x^3+4 x^2+16 x+3$
• $y^2=x^6+12 x^5+13 x^4+19 x^3+39 x^2+9 x+11$
• $y^2=30 x^6+3 x^5+33 x^4+3 x^3+36 x^2+28 x+31$
• $y^2=19 x^6+27 x^5+8 x^4+33 x^3+26 x^2+x+13$
• $y^2=38 x^6+34 x^5+11 x^4+17 x^3+27 x^2+10 x+40$
• $y^2=7 x^6+27 x^5+x^4+16 x^3+10 x^2+10 x+10$
• $y^2=x^6+38 x^4+27 x^3+25 x^2+24 x+6$
• $y^2=27 x^6+25 x^5+18 x^4+8 x^3+4 x^2+39 x+21$
• and

• $y^2=26 x^6+27 x^5+8 x^4+18 x^3+7 x^2+40 x+3$
• $y^2=8 x^6+16 x^5+28 x^4+10 x^3+10 x^2+13 x+24$
• $y^2=38 x^6+23 x^4+10 x^3+31 x^2+23 x+24$
• $y^2=3 x^6+5 x^5+32 x^4+28 x^3+29 x^2+28 x+17$
• $y^2=21 x^6+39 x^5+39 x^4+25 x^3+23 x^2+37 x+22$
• $y^2=25 x^6+35 x^5+18 x^4+17 x^3+16 x^2+26 x+34$
• $y^2=4 x^6+15 x^5+x^4+7 x^3+x^2+29 x+32$
• $y^2=32 x^6+10 x^5+6 x^4+27 x^3+20 x^2+40 x+13$
• $y^2=9 x^6+3 x^5+5 x^4+24 x^3+13 x^2+12 x+26$
• $y^2=24 x^6+34 x^5+4 x^4+22 x^3+29 x^2+6 x+18$
• $y^2=19 x^6+30 x^5+19 x^4+9 x^3+37 x^2+2 x+35$
• $y^2=14 x^6+13 x^5+21 x^3+20 x^2+24 x+3$

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 4.0.382965568.1.

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.g_y	$2$	(not in LMFDB)