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

• Label: 2.41.af_dh
• 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:	$1 - 5 x + 85 x^{2} - 205 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.390929052305$, $\pm0.482661339481$
Angle rank:	$2$ (numerical)
Number field:	4.0.4020341.1
Galois group:	$D_{4}$
Jacobians:	$23$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1557$	$3078189$	$4787153757$	$7973768489301$	$13419519962188752$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$1827$	$69457$	$2821811$	$115829102$	$4750170507$	$194755199417$	$7984925310403$	$327381915269317$	$13422659285963502$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 23 curves (of which all are hyperelliptic):

• $y^2=25 x^6+17 x^5+33 x^4+32 x^3+19 x^2+6 x+19$
• $y^2=35 x^6+34 x^5+34 x^4+23 x^3+21 x^2+12 x+17$
• $y^2=10 x^6+5 x^5+15 x^4+38 x^3+25 x^2+13 x+31$
• $y^2=40 x^6+11 x^5+23 x^4+37 x^3+17 x^2+23 x+7$
• $y^2=40 x^6+10 x^5+39 x^4+15 x^3+39 x^2+24 x+20$
• $y^2=38 x^6+24 x^5+33 x^4+34 x^3+26 x^2+33 x+34$
• $y^2=20 x^6+24 x^5+21 x^4+16 x^3+12 x^2+38 x+27$
• $y^2=14 x^6+21 x^5+34 x^4+38 x^3+20 x^2+31 x+3$
• $y^2=24 x^6+4 x^5+31 x^4+2 x^3+13 x^2+22 x+8$
• $y^2=26 x^6+35 x^5+10 x^4+12 x^3+31 x^2+40 x+24$
• $y^2=17 x^6+33 x^5+30 x^4+34 x^3+3 x^2+31 x+40$
• $y^2=6 x^6+28 x^5+5 x^4+9 x^3+40 x^2+18 x+28$
• $y^2=31 x^6+18 x^5+32 x^4+4 x^3+3 x^2+24$
• $y^2=34 x^6+9 x^5+8 x^4+26 x^3+5 x^2+12 x+30$
• $y^2=4 x^6+29 x^5+2 x^4+10 x^3+9 x^2+39 x+4$
• $y^2=28 x^6+27 x^5+19 x^4+x^3+39 x^2+x+21$
• $y^2=30 x^6+19 x^5+27 x^4+25 x^3+36 x^2+25 x+40$
• $y^2=20 x^6+12 x^5+34 x^4+7 x^3+25 x^2+2 x+1$
• $y^2=16 x^6+40 x^5+20 x^4+40 x^3+33 x^2+8 x+33$
• $y^2=3 x^6+8 x^5+3 x^4+18 x^3+32 x^2+8 x+37$
• $y^2=26 x^6+25 x^5+30 x^4+23 x^3+40 x^2+10 x+34$
• $y^2=21 x^6+11 x^5+17 x^4+30 x^3+x^2+4 x+40$
• $y^2=37 x^6+3 x^5+3 x^4+16 x^3+8 x^2+x+21$

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 4.0.4020341.1.

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.f_dh	$2$	(not in LMFDB)